How to find "doorskin"?
I'm surprised that you can't find 1/8" plywood (luan), but have you asked about Thermo-ply? It's a thick waxed cardboard, supposedly for sheathing (ha!) but suitable for your application. 

Ramboard (somewhat thick cardboard, sold on the roll) would be suitable if you used entire sheets (as opposed to strips). The factory edge on this is handy for straight lines.

Also possible (but a pain to cut into strips) would be countertop laminate material... a good lumberyard would probably have some old/kicked around product for cheap. (But then, if you had a good lumberyard, you'd already have some 1/8 luan...)