I do not think the Blickman style false bottom will be any improvement compared to the screen you are presently using. In fact, I think that it is a move in the wrong direction. Using a screen, you essentially have the best ability to let wort flow evenly thorugh your grain bed. The Blichman is pretty close to providing that, but the design is geared slightly more toward longevity and easier clean up. It can support a larger grain bed, but as long as your screen is not warping, the more robust desing will not be a benefit.
Things that I have noticed that have reduced my mashes from sticking while sparging:
1.) Stir the Grist and get all the pieces in suspension before letting the grain bed settle. The larger pieces will naturally fall faster than the small, reducing the small bits from clogging your screen. To aid this I add the water FIRST, then add the grains.
2.) Open the valves 100% every once in a while. I do not have a pump system, but I did notice that grains would sneak through and sometimes build up in front of my valve that is metering flow. By opening 100% and then back down to the metering position, the grain bits get unstuck and normal flow continues.
3.) Grind the Grain a bit bigger/add rice hulls. Depending on the mill, your husks may be getting chewed up a bit too much by your mill, and not providing the "large" filter pieces needed to prevent a stuck sparge. If you love your mill setting and do not want to change it, add some rice hulls to your grist (.25 to 1lb of hulls depending on the amount of grist) That will supply extra "large" pieces without forceing a change to the milling size.
4.) Cut cross channels in your grain bed during sparging... about 1/3 to 1/2 grain be depth. I have noted that some malts are high in gummy proteins that really slow down sparging if not addressed. One way to reduce the gummy proteins is to do a protein rest around 130F. However, an easier way to address the issue is to cut cross hatches into the grain bed, about 1.5 in square into the grain bed to break up the protein gunk and allow flow. One key identifier of the issue is a lot of grey sludge floating in the grain bed about 10 to 15 minutes after dumping in the grains. NOTE: it is important to not cut the bed too deply as that will lead to channeling and reduce your efficiency. It is also important to cut uniformly for the same reason.