Cyclocross Bike Build Advice?

Your budget and expectations are not realistic. It is never cheaper to build a bike from parts unless you already have most of the parts you need on hand. Bullhorn handlebars are very limited on any bike, drop bars are much more versatile. You might pull this off if you had years of experience with bikes, but as a newcomer to this you will not come anywhere near to putting a bike together anywhere close to your budget. Sell the frame and buy a bike