If you already have 1 inch thick strapping over your concrete, then I'd probably just use 1 1/4 inch long drywall screws to screw the drywall to that wooden strapping.
EXTRUDED foam insulation is an advantage in my view because extruded foam insulation like the blue Roofmate insulation doesn't need a vapour barrier. Even expanded foam insulation is pretty impermeable to air leakage through it, but it's not right to put it in without a vapour barrier.
There are two kinds of polystyrene foam insulation; extruded and expanded. Extruded is the (typically) blue or pink styrofoam insulation you see around. Expanded is the white stuff that looks like it's made of small "beads". The difference is that they use more blowing gas in making the expanded styrofoam insulation, so the "beads" of styrofoam can be interconnected, and that means that expanded styrofoam can be permeable to air, and that means that condensation can get trapped inside it.
Extruded polystyrene foam insulation, on the other hand, uses less blowing gas when it's made, and so the tiny bubbles that form inside it won't be large enough to be interconnected, and that makes it effectively impermeable to air (humid or not). Since humid air can't get into extruded polystyrene insulation, you don't need to put up any vapour barrier to keep humid air out of extruded polystyrene insulation.
So, you just cut your extruded polystyrene insulation to fit in the spaces between your strapping and use that expanded foam stuff to seal any gaps between the insulation and the strapping, and hold the insulation in place at the same time. When the expanded foam is cured, just cut it off flush with the front of the strapping. Then put up your drywall.
Extruded and expanded polystyrene foam both have an R value of about 5 to 5.5 per inch of thickness compared to about 3.5 per inch of thickness for fiberglass. My understanding is that extruded foam has a little higher R value per inch than expanded, but the difference isn't very large; maybe an extra 0.25 to 0.5 maybe.
Extruded is appreciably more expensive than expanded, but you can't with good faith put up expanded without using a vapour barrier. They use expanded polystyrene foam to make parts for fridges and freezers, and it's not uncommon for these parts to become water logged due to water accumulating inside the foam.
Also, pay no attention to extruded or expanded foam insulation advertisements that claim very high R values for their foam, like R 8 to R13 per inch. It's very possible that the INITIAL R value of foam insulation when it's first manufactured might be as high as R8 or R9. But, as the blowing gas in the foam escapes and is replaced by air, the "stabilized" R value diminishes down to about 5 to 5.5 per inch. But, since the people selling R9 foam insulation are lying to you already, they see no harm in promising you an R value of R13 just to sell more insulation than the guys trying to sell R9 foam insulation.
Stay away from anyone claiming their foam insulation has an R value over 6.5 per inch because that's about the upper limit even with polyisocyanurate foam insulation with a metal foil radiant barrier. Anything over that is bogus.