That mesh doesn't look like its plastic coated, I think it is just plain old woven wire cloth metal material judging by the photo - in the past, I have spoken with manufacturers on this stuff - www.bwire.com - and they always say to use aviation snips on this material. As far as the drywall...
Instad of plywood how about putting over some heavier wire mesh to reinforce the weak stuff? That is probably just some cheap stuff from over seas (obviously), but wire mesh does hold some advantage ove plywood here. For instance, wire mesh can be see through, this way you can see whats up on...