Cross tires on a road bike
The main thing to watch out for is the brakes, if the comes with V-Brakes or Cantilever brakes, you should be able to run wider tires and fenders.

Typical CX tires are 32-35mm and typical road from 23-28.

The caliper style brakes found on many road bikes will not accommodate the wider CX tires. There are many options under 2-3K, You could even look at cyclocross bikes as many are under the 2-3K in the US. Visit your local bike shop and look at a few options.

A buddy bought a Kona "Jake The Snake" last year and is a very nice bike and under the 2-3K.