What belt-drive compatible frame and belt-drive kit would you recommend?
You may be a bit out of luck with that price range if you hope to get the whole kit.  The Gates belt, cog, and "chain"-ring retail online for over $200 by themselves.

That said, there are some options if you are clever and dedicated. If you're able to find one of the old 90's mountain bikes with elevated chainstays you can use one of the expensive ($125-$150) eccentric bottom brackets that mount in a normal 68mm shell for tension.  A 90's Nishiki should be fairly cheap, so you'd eke in just under your total price.  It's also fair to say that your bike would be a unique build.

Alternatively, if you have access to an affordable machinist, start with an inexpensive frame that has thick track-style dropouts. Modifying them into a variant of the old Spot-style dropouts should be relatively easy with a dremel, some taps, and a skilled hand. This avoids the significant cost of the S&amp;S couplers used by most belt-drive modifications.

Unfortunately, unless the frame was purpose-built for a belt, you're going to be stuck modifying it. There are few, if-any, belt-specific frames in the market now, which keeps prices high.  The proper modifications either require an expensive S&amp;S coupler or new dropouts, which are expensive to add aftermarket.