For what you have in mind, as long as the steerer matches the head tube, you'll be OK, the fork rake is not massively different. A CX fork will have blades about 40-50 mm longer, so if you fit a road fork, the front of your bike will end up a bit lower, but it's a fixed bike, so I assume you don't care too much... on the up side you'll be more aerodynamic...
For what you have in mind, as long as the steerer matches the head tube, you'll be OK, the fork rake is not massively different. A CX fork will have blades about 40-50 mm longer, so if you fit a road fork, the front of your bike will end up a bit lower, but it's a fixed bike, so I assume you don't care too much... on the up side you'll be more aerodynamic...