What types of bikes are you looking for?

If it's a standard 26" MTB, there are plenty with discs to choose from, and since they also use a 135mm rear spacing, the Alfine hub would fit no problem. All you'd have to do is move the rotor from the original wheel to the Alfine wheel and you'd be in business as far as the brakes go. The Alfine does take a different shifter though, incompatible with derailleur shifters, so that would also have to be swapped.

If you're looking for hybrids with 700c wheels, you'll have less choices for models with disc brakes, but they are out there and will certainly accommodate fenders and racks. You may have a tougher time finding a prebuilt wheel with an Alfine hub (cheaper to buy prebuilt than have a custom wheel built for you), but again it's not insurmountable. You could also trade in the shifter and rear derailleur for credit if the shop allows that (and if the bike has them).

Were it me specifically in your situation (and we are near the same size), I'd be looking at a 29er MTB, perhaps even a singlespeed like a Raleigh XXIX, a Redline Monocog Flight or a Kona Unit 2-9. The large wheels just sing on the pavement with smooth tires, and if you went down in tire size a little, fenders should fit too. You'd still have the issue of finding an Alfine wheel, but again, a decent shop will work with you and it could all be done for under your budget.

Keep in mind that with any of these bike types, you'll need a disc-compatible rear rack...or a shop that knows how to modify one for discs. I am not sure if any of the models I listed have the required rack mounts, but even that isn't a dealbreaker. Racks can be adapted with 'P' clamps to almost any frame. As for the compatibility with your trailer...at least you have a few more options to check out.

Best of luck in your search.