Sounds like you are thinking of something like this. It's an early Mazda pickup chassis I experimented with about 5-6 years ago to see if I could do what you are talking about. I think these two photos will help point out some of the pros and cons of going this route.
If you look closely you'll notice that the frame had to be highly modified just to get it to look this good (bad). In their stock form, mini pickup frames are too wide and too long to accommodate a typical 20's or 30's type body (even if that body style is highly modified). They also have lots of curves and humps built into the side rails of the frame. What I did was cut the Mazda frame just ahead of where it begins to curve up over the rear axle and then cut off the entire front cross-member section. I then welded the Mazda front and rear sections back together with 2x4 tubing. This allowed me to narrow and shorten the frame to fit a 30's type body. And for me, at least, that was about the extent of the "pros".
On the "con" side of the ledger here are a few of the most notable drawbacks:
1) Most of these older pickups have highly rusted frames (especially the ones from the Midwest where JerryMopar happens to live). Used in their stock form, this makes for a potentially dangerous ride and more importantly, these highly rusted frames are very difficult to weld - either to themselves or to new material introduced for strength and support.
2) As you can see from the second picture, this is a very cluttered and bulky looking front end. Not something that will look real good without full fenders. And worse, the idea of full fenders in nearly impossible because of the shock/spring tower height and size. You would have to make totally customized fenders, and even then I'm not sure you could make them fit and look right.
3) Yes, you certainly COULD cut off the entire front cross member and suspension set up and replace it with a traditional straight axle, but then you are into as much or more cost than just building the entire frame from 2x3 square tubing. You are going to have to buy all the front suspension parts anyhow (axle, springs, shocks, radius rods). I found in building the rat project that the cost of the frame itself was relatively small. It is the suspension parts which really add up. (See budget breakdown in my journal.) So if you can't use the suspension from the mini pickup donor, then you don't save much money.
Don't get me wrong. I'm not saying it can't be done. Mini pickups have been used very effectively under certain full bodied, full fendered cars. But my experience building low budget 20's - 30's type rods is that scratch building the chassis turns out to be about the same cost as trying to modify a mini-pickup frame. And the "look" of a scratch built, traditional chassis, is much better in my opinion.