Yes, it can be done that way, but not the other way (not Briggs to Murray). I converted a 1929 Murray Town Sedan interior to a 1930 Standard Briggs Fordor Sedan 15 years ago. The only thing you'll have to do is trim the rear door panels narrower because unlike Murray sedans, the Briggs rear doors are not the same width as the front doors. Fortunately, because the Murray doors are all the same width, you'll have extra cardboard and material that can be trimmed to fit the Briggs rear doors. You might have to peel the material edges back on the front panels and add a strip of cardboard to each end to make the panel a bit wider. I don't recall doing this, but be prepared to do that. There's usually enough material stretched over to the back side that will allow it to be used to cover the added one inch needed. It may be necessary to slightly modify some trim panels above the windows beneath the headliner, but once again, there should be plenty of material on the back side to do this.
As for the top liner, I think I had to order a new one from Lebaron Bonney because the top bow ribs were spaced differently so that the cloth strips that secure the material to these bows didn't match. But if your wife can sew, she can alter the strips to match the top bows. If absolutely necessary, though, you could order just the top lining to match your bows.
The interior turned out nicely otherwise. You'd never guess it had once lived in a Murray body. By making these slight modifications, you will save yourself thousands of dollars in a new interior.
Marshall