A client is being difficult about paying my invoice from September 2024 - England

I conducted some freelance construction work from 17 Sep 2024 to 21 Sep 2024 and never sent my invoice to the client. Back in June this year I messaged one of the colleagues from that time and he told me I was good to send my invoice, as I needed the money. I sent it on June 4th, stating that the payment deadline would be in 7 days, so June 11th. I received no response until June 25th, where the client explicitly told me the invoice had been approved for payrun at the end of the month. Come the end of June, I had received no payment and no follow up emails explaining the delay. I sent more emails, before finally sending a Letter Before Action on July 2nd, which stated that I had to be paid within two weeks, so July 16th, or I would be forced to take the matter to a small claims court. I finally received an email from the client on July 16th, who was now saying that there were issues with my invoice:

1. That I listed an extra day
2. That some hours were inflated
3. That the client had chosen to "entertain" the invoice in good faith, despite it being submitted nine months late
4. That my tallied hours do not meet the total In short, they refused to pay.

I amended the invoice, which were typos at best, but I do understand that it is a legal document and so I happily fixed everything. I removed the extra day and consulted the rota from the work schedule, readding up the hours. I need to point out that while the hours have been inflated by 30 mins to an hour, I have photo prood on my phone that I was basically doing overtime as I was working with date, location and time stamps. I sent the amended invoice on July 16th, stating that it had to be paid by July 23rd. Of course, July 23rd rolls around and there is no payment and no email explaining why. I felt close to giving up at this point. July 24th I send a gentle reminder that I amended the invoice and that the payment is overdue, with no response. I sent another one a day or two later. No response.

I sent another email today, August 4th, asking to be paid, that I'd double checked the invoice and that I just wanted the matter to be resolved. The client send an email a few hours later, that my hours are still inflated and that I need to remove the VAT. I replied, asking the client to specifically tell me which hours I was logged as working, because what I have written down is from the work rota. I am practically begging at this point, but the client is yet to respond. Over the course of two months, I have received a total of 3 emails from this person. I don't know if what the client is doing is unlawful in any way. I am just a university student who is trying to get paid and the client seems to be dragging this out for as long as possible until I give up. Keep in mind, the payment I am asking for is 480 pounds and the client is a company who is well within their means to pay this amount of money. Any advice is welcomed, I am at my wits end here.