You can make whatever kind of deal to which you can get the dealer to agree. I have done all sorts of weird deals over the years. Probably harder now with the car shortage though; dealers have the power at the moment.
If you tell the dealer this is your plan, they may try to screw you by lowering their offer 2k.
I would either tell the dealer you don't plan to put anything down from the trade (have them just cut you the 10k check) and then just spring it on them later, or sell the car to another party first.