"Maximum" down payment allowed?

I test drove two trims of a vehicle today that I really liked; I've been shopping online for a few months and have been keeping an eye on manufacturer incentives and rebates, and am also eligible for a partner program through my employer.

I told the salesperson I was eligible for the partner program but understood that sometimes the dealer had better offers, especially since the particular vehicle I was looking at was an outgoing model, so I wanted to get a better idea of what the best option would be for me financially, and I was honestly torn between the two trims/upgrades. After "going to talk to the owner", he came back and showed me some numbers that included a higher APR than the current advertised incentive rate with the added caveat that I could only put a maximum of a $2,000 down payment on the deal. He said that with the discounts the dealership was offering, I couldn't get the APR rate as well, so I'd have to have the higher APR, but there wouldn't be any kind of prepayment penalty.

The whole thing seemed a little off, and I told them I'd need some time to figure out which trim I'd rather pursue, (which was absolutely true). I was really turned off by how high the monthly payments would be, and I'd rather put a much higher chunk down, but was told it wasn't an option. (I was told that I could, however, put a lump sum payment into the loan after it closed, but that wouldn't lower monthly payments, just pay it off quicker).

Now that I've had a bit to think things through though, this was the first time I'd ever encountered any mention of a maximum down payment for financing--is this a common condition when qualifying for special financing rates?

I am capable of paying in cash, am I better off waiting until there's a larger cash incentive as opposed to a really good financing rate? I was willing to pay some interest for the sake of cash liquidity over the life of the loan, but the higher rate combined with the limit of $2,000 on a down payment really took me aback, and now I'm wondering what my best course of action will be.

TL;DR - liked a car, but dealer wouldn't combine discounts with published incentive APR rate, then insisted that I could only put a maximum of $2,000 down to qualify for their "special" deal the owner was willing to make me, which included a higher APR rate. Are maximum down payments really a thing?