I have recently had some problems with my false bottom and want to experiment with a mash screen. This might be a bit of a stupid question but I have never used one before: Can you do the mashing with the screen already mounted or do you have to fill the mash onto the screen in a second tun? My worries is that when you are heating the mash, the mash underneath the screen won't be moved and hence possibly burn.
If you have a false bottom you may want to try this out if you are getting a screen.
This method helps to improve the filterability of the false bottom. THis way you can mount it all at once, mash on top and not have to move to a laughter tun.
You should be able to rig something up to mash and lauter in the same vessel.
EDIT: based on new info from question/comments.
SO NO a bazooka screen can't be used in a direct fired mash tun set up. Its very likely you'll burn the mash, unless you stir like crazy.
I'd recommend you try mounting a bazooka screen under your false bottom. The FB will give you large particle filtration and the Bazooka will take care of smaller particles that make it under the FB.
If you are really having a problem with the false bottom and stuck sparges, I'd recommend looking at your crush. The crush might be too fine and you are getting some clogging. A second approach would be to toss in a half pound or so of rice hulls to improve the lauterin efficiency of the grain bed itself.