Well, if I'm understanding you correctly, you're saying you won't need any 120v ac power supply?
As long as you don't need alternating current, you'd be OK. Would you run your fridge on LP all the time? No reason you can't, just asking. What about a microwave? You could rectify 12v to give you 120v, but batteries wouldn't last long trying to run a microwave or hair dryer.
There are newer, better converters available. Progressive Dynamics makes some pretty nice ones that a lot of folks on here use (I have one myself, just haven't installed it yet, but I'm converting my 34 footer to 50amp). Xantrex (sp?) makes some nice stuff too. And I'm sure there are others.
But, if you don't need 120v and can get by fine without air conditioning, microwaves, or hair dryers (unless you really do a neato torpedo 12v to 120v rectification), you could do this.
The really old trailers didn't have 120v. They got by in the 30's without it. No reason you couldn't today.
As for me, my wife really likes 120V for her hair dryer and I like air conditioning
Best of luck,