Tony, there's a guy in Corvallis, Oregon named Paul Ede who's quite an authority on mandolin-banjos. I happened to read an article he'd written about them in a now-defunct mandolin magazine, and he showed that the key to getting them to stay and play in tune is to use the sort of bridge you'd use on a flat top mandolin, like this one:
Ede recommends using an all-maple bridge in more or less the same shape as this one, with ultra-light strings and a straight, non-intonated saddle area.
My repairman ran one of those up for me, and I strung my mandolin-banjo with the same super light gauges Ede recommended. While it was very loud, and also a vast improvement over the stock banjo-style bridge, I felt we could do better, both in terms of tone and intonation.
So we went up to a heavier gauge of strings (still a light gauge by mandolin standards,) and my repairman made up an ebony bridge with an intonated bone piece on top. It was similar to but not identical to this Gold Tone octave mandolin bridge:
That was much better, but it darkened the tone a bit more than I wanted, and it wasn't nearly as loud. So what I had my repairman do is cut off the bone and ebony part at the midpoint, and he glued that onto a wide maple base that's close to five inches long.
It's counter-intuitive, but having that wide base seems to transmit the sound better on the banjo. It's incredibly loud, but having the ebony seems to sweeten the tone quite a bit. That combination - intonated bone top, ebony middle, and wide maple base, all glued together - gave us the best tone and best intonation by far.
You might also want to experiment with plastic and/or Fiberskyn heads on the instrument. My first mandolin-banjo had a skin head, and it was a constant challenge to keep the tension where it needed to be. The plastic head is much simpler - once it's fully stretched in, it stays where you put it.
So those are my suggestions: forget about trying to make the traditional style banjo bridges work on a mandolin-banjo, because on this instrument they're a source of constant aggravation, and simply won't stay in tune. Get or make a flat top mandolin bridge instead, and experiment with the sort of layered construction that my repairman and I figured out over the course of working with this.
And get a plastic head of some sort. Not only will it be far more stable than a skin head, but you can vary the tone colors as you play quite a bit more, as well.
Hope that makes sense.
Wade Hampton Miller