FYI. My current bike uses a 36volt lithium ion battery from elifebike on ebay that cost $280 shipped. It also uses a 36 volt 500W geared rear hub motor, also from ebay for $199. I've got 250 miles on it so far, and top speed without pedalling is 18-20 mph. Runs fine. Very reliable so far in the four months I've had it running.

I like geared motors because there's no drag when pedalling. I bought a second 36volt motor, same as the first, and will be using it as the front motor in a second bike I'm building. Wheel and hub weighs 12 pounds, by the way.

I would go with a brushless motor. Controllers are more complicated, but they're all mass produced these days with vendors copying each other's designs, so the electronics are pretty cheap. A motor with brushes will eventually wear out the brushes, no?

So, yes, I think your budget is realistic with a $249 trike, but I think 20 mph is not. These trikes were intended to be pedaled well under 10 mph by geezers. They are not stable when turning at speeds. The brakes will probably be an adventure above 10 mph too.