Installing 1inch foam board in ¾ inch gap under drywall

[wallnot]

Hello Everyone, nice forum here. I,m a homeowner fixer-upper.
My corner bedroom has drywall on 1x4 studs – 3/4” gap between concrete block and drywall.
There is no insulation and its very cold in winter. I want to rip out the drywall and install 1” foam boards between the studs. To do this I would have to increase the gap space from 3/4” to 1 1/8”.
I”m considering one of three options.
A sheet of 1/2” (a) sheet rock or (b) plywood, rip 3 1/2” strip and tack them on top of existing 1x4 studs.
I prefer the sheet rock – its cheaper and easier, but will it hold up.
The third way is to use 2x3 re-saw to 1 1/8” thickness. That would have more “screw-bite” but would be more costly and a lot more work involve plus removing the old studs.
What do you guys think?
Thanks
Wallnot

[TonyM]

Just use a screw of the appropriate length as not to go through to the block. Alternatively, use insulated plasterboard, that way you would also avoid 'cold bridging' at the battens. You could just put insulated plasterboard over the existing drywall.