If I sell my old car privately instead of trading in, is it safe to assume the total will only change by the price of the trade in plus the additional tax I need to pay?

Hi everyone, I'm in the US (Pennsylvania), and I am waiting on a BMW M240i Xdrive that I ordered a few months ago and it's almost time for the car to arrive at the dealership, so I've been exploring options for financing a bit more.

The sheet I was given shows:

Market Value Selling Price $65,150.00

Total Savings $4,007.50

Adjusted Price $61,142.50

Trade Allowance $16,500.00

Trade Difference $44,642.00

DealerDoc $490.00

Tax $2,678.55

Non Tax Fees $223.70

Balance $48,034.75

If I sell my car privately, is it safe to assume the numbers would be as follows?
...

Adjusted Price $61,142.50

Trade Allowance $0

Trade Difference $61,142.50

DealerDoc $490.00

Tax $3668.55 // PA sales tax on new balance of $61,142

Non Tax Fees $223.70

Balance $65,524.75

(assuming 19,500 from private sale): Balance is now $46,024.75

So, private sale nets me an additional savings of $2k.

Carvana quoted me around 19,500 for my car last I checked, so even with the loss of the tax credit for the trade in, this still appears to be the better deal, right? I guess the crux of my question is, is the dealer likely to want to play with that adjusted price / "savings" if I am not trading in a vehicle, e.g if trade in margins were factored into that discount they applied to the MSRP?

I will be financing the balance after my trade in and after a $20K down payment.

Thanks