Most of the year my cyclocross bike does the duty of base mile bike/rain bike. I swap to a bigger chain ring for road and obviously change tires, and put on fenders. Using the whole bike vs. having a complete separate frame seems easier to me.
The entire drivetrain of a cross bike is from the road bike aside from cross bikes generally are geared down for the large chainring, but the seat tube may have a different diameter so you may need a separate seatpost and front derailluer clamp. Just seems like a lot of work with out much gain when a cross bike with road tires works fine for road training.
I was in your position earlier this year. I opted to move over the parts that I could and make my 'cross bike my primary road/training bike. I just went to a 50T large chainring and left everything else the same. If you race on the road (I don't) then this may not work for you. But if you need a budget approach to a "do it all" machine, I'd leave the cross bike built as your primary training rig.