Need advice about a client not wanting to pay after services were already completed

Here’s the texts: https://imgur.com/a/e3oAcID

Background: I am a professional organizer, my company is new, but I have been doing direct and adjacent work since college. I have an extremely diverse background that makes me highly sought after in my area - my rates are (****very transparent.***)

To fill gaps while getting my company up and running, I reactivated my TaskRabbit profile. I offer a handful of organizing related services, including move related services.
Since I am on TR as a way to meet potential clients and get some basic, one day projects done, my rates on TR are a bit lower than my usually rates… (***Despite this, they are high for TR and I have been getting hired multiple times per week.***)

Along with the fun day jobs, I occasionally accept project requests on TR that are a closer match to what I do professionally.

I talk about my background, experience, skills and company on my profile. I also link the website, and list a handful of specialty services with the statement “For a quote on Speciality Services, please attach photos”

Yes I know that’s not how Taskrabbit is supposed to be used, but it’s worked out thus far.

(***MOST OF THE TIME***), clients will ask that I invoice them directly. I am direct in letting them (multi day project requests) know that (***the rate they agreed to on TR and my off app rate are the same, with a few exceptions:***)
**It is a hoarding house -** I am equipped to handle moderate hoarding situations, but it requires a lot of extra care and planning. (Not at the level of extreme hoarders, ask if you want me to explain)
**The client needs the project completed immediately/under a time constraint - **I have a higher rate for work that needs to be done in under 48 hours.
**The Client request that I work outside of normal hours -** This includes anything after dark or before sunrise.

Anyways, last week, as I was wrapping up a 6 hour day project, a client hired me, asking if I could start (***that day***) after 7PM, because he needed help (***”Unpacking boxes and putting clothes in closets and kitchen stuff in kitchen cupboards”***)

I asked if I could come see the situation, how many boxes, how much space, etc etc. He agreed.

The project was enormous - Over 100 boxes, delicate fine china, 2 households being combined, kitchen 100% blocked floor to ceiling with full boxes. On top of that, what he actually wanted me to do was (***independently, unpack, organize, and set up a 2 story condo he’d JUST moved into, before his elderly mother moved in 2 days later.***)

He said word for word “Please figure out what you would like me to pay you.”
I told him a few important things:
I actually could handle a project this size. I said I would start independently, but would need help from a colleague the next day(business partner). He agreed to hiring a 2nd person.
This was not a normal unpacking job. It was a complete move in.
I told him that due to the scale of the project and the timeline, my rates are higher, starting at $120 an hour.
He asked how much I thought it would be in the end, and I said “let me start working first, at $120 an hour + BP at $60/hr, I won’t surprise you with over a $6,000 bill.” This is the 2nd time I told him the rates.

I worked alone until midnight, when he told me he was going to sleep and asked if I was able to continue until “I got too tired” and to take the boxes/packing materials when I left. I agreed.
At this point, without me asking, Client told me he’d be happy to pay for hauling the trash.

I worked until around 4 am. I did so because he had also texted to ask me if I could work overnight, and I had agreed in writing. I had also told him my hourly rate twice, and he had nodded his head and verbally agreed with “ok.”I took this to mean we had an agreement.

Same thing next day, but with BP.

BP shows up 2 hours before me at 3pm, I arrive at 5pm (after working with a different client).
(***At this point, I realize a few things:
Client has 5+ of every single kitchen tool.
Client has over 50 cups
Client has over 80 plates
Client has over 60 bowls
Client has **AT LEAST** 12 teapots
Client has 2 or more of every large kitchen appliance (including microwaves, toaster ovens, crock pots, rice cookers, cuisinarts - both large and smaller “electric choppers”)
Client wants to “save everything” because his “nieces and nephews will want it” 🙄
Every inch of storage space is maxed out, including 6 china cabinets, and the tv stand, which is filled with cups.
Point being: We are dealing with “Organized Hoarding” behavior. I don’t mention it, but I remind client that I’d be happy to plan some time to go through everything and decide how to store it, maybe sell some of it, donate some so it doesn’t end up in the trash, etc.

I mention this because the fact that it is a hoard once again changes the project scope.

Anyways, we work until 11pm. At 11pm, same request from client - “can you keep working a few more hours?” We agree, and end up leaving again with 2 SUVs fulls of packing trash.

The next day, same thing.
We start at 2pm.
Around 5:30, Client is leaving to go pick up his mom and see some family. He mentions again “please figure out what you would like me to pay you.”

This concerned me slightly, but we’d talked about it multiple times, so(***I again remind him that due to the project scale, including the abnormal hours, physical labor, and time constraint, my rate is higher than my standard rate, and repeat that it is $120 an hour.”***)

He just says thank you, that he is leaving to go see family and pick his mom up, and asks that we leave by 10.

Around 9:45, him and his mother return. We leave by 10, he thanks us again and asks me to please invoice him as soon as possible so he can pay us for the work up until then (the plan at this point was for BP and I to wait about a week, so his mom could settle in, then come back to finish the job)

Flash forward to today, 2 days later.
I’m working with a new client (amazing woman, super laid back, anyways, total opposite…. BUT SHE HAS A LOT OF CUPS) sorry to rant - I spend like 3 hours figuring out what to do with the damn cups, get frustrated, and at the same time, receive a string of back to back to back calls that just won’t stop buzzing and buzzing.

I decide to take a break and go to the bathroom. I look at my phone, and there’s over a dozen missed calls from a contact that I mistake for Client (same first 3 letters). I’m already stressed, I have to get back to work, so without thinking about it I copy and paste an old draft I had of Clients Invoice.

He responds (as you can see in the texts)… I panic. I don’t think about the rate we agreed on, I don’t think about having worked overnight, or the hoard, or the physical labor, nothing.

I go into panic mode. I know my texts read as calm/collected/accepting, but I was actually hyperventilating in the bathroom. (I like to think I sound professional so maybe that’s it)

After finishing up for the day with my new client, going home, eating something, and sitting down the reread the texts from Client 1, I feel massively disrespected and cheated.

Yes, I sent the first text. But it was under duress and was an accident. I doubt he would have asked a moving company or an in home nurse the same question.
Idk if y’all can tell from follow up message, but I wasn’t thinking straight… I would NEVER say something like “That would be at a rate lower even than what | would have gotten from TaskRabbit after they take their large fees out of the posted rate.” In a normal situation.

Also, the “Ok!” Response feels a little to excited/happy.

Anyways,
I’m still freaking out.
The labor is completed.
The rate was agreed upon.
I repeated it multiple times.
The circumstances surrounding the task were unusual and above normal service standards.

Am I justified in sending the actual, formal invoice that isn’t just a text, saying something like (***”That initial text was never ment to be sent. The responses that followed were reflexive, as well as being the result of complete shock. know that you have already sent me part of the agreed upon amount. I will send an invoice for the correct total, including a subtraction of the amount already paid.”***)

Do I have to explain anything more? I feel completely manipulated, ESPECIALLY considering I sacrificed sleep under the agreement that I was working at a premium rate.

Any input is really appreciated.