Hanging drywall: How to shim 1.5"?
My thoughts the same about drywall @Comintern, to make up the 1.5 it would be easiest to attach 2x4's sideways.
If this is now a inside room I would remove the old exterior siding, However the “cement shingles”  look like the asbestos type commonly used you might want to have them tested prior to working with them for health reasons.
I don’t think I would ever put drywall over the lap siding even with the sheet insulation if now an interior room. If this is still exposed to the elements you really should use exterior sheeting.