IMHO,
There will be no pre-payment penalty on whatever the dealer wants you to do to take the car that day (double check to be certain). You can do whatever loan they want to do at 100% financing, then a month later or less re-finance with your credit union and do your downpayment there.
Alternatively you can wait for the check to clear as of now and make sure to mention you will not take the car if it has another mile on it when you come back since they are forcing you to wait. Take a picture of the odometer and wait another few days.
I think they are wanting the kickback percentage from getting the loan through them and are butthurt you are lending through someone else. I could be wrong but that's what it seems like here.
My next time around I'm planning to not my any money down and finance the entire thing over a longer term to keep payments down. While I can pay for it outright, I personally like the flexibility of keeping money on-hand for any unforeseen expenses related to my house, etc. For me, that interest rate I'm paying is well spent, since I don't have to take it out of my savings account. I know long-term that's not the most cost-beneficial solution, but in the end, assuming I financed, say, $35k, we're talking around $3500 in interest over those 5 years, or about $700 per year. For me, if $700 broke me, I wouldn't be looking at this trim/car.
I personally do not understand why anyone pays cash for a car. If you have that money in savings you should easily be able to find a money market account or even just relatively secure index funds that are going to at least equal if not beat your interest.
So say you got $50k to pay cash on a car. You can dump all that money in a depreciating asset and have no monthly payment or:
Get a loan @ 4% for 72 months and pay $782/month $56304 total
and
take your $50k and put it to work at the same simple interest at 5% and over the same 72 months you will have paid off your car and have $67,500 in that account or more likely you're getting compound interest at 5% so over the same 72 months you will have paid off your car and have $70,355
At any time you feel uncomfortable or something comes up you can always sell your car or pay it off from the funds in the account. Even selling at a loss you will have more net money.
To each their own and I get the safety/security feeling of not having a payment and having something paid off but it's not the best financial path.