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Roll of 2mm EVA foam just arrived from ebay and looks like the business. Thanks for the recommendation. This might be flexi enough to sew under the liner while still firm enough to hold shape. Weekend project ahoy!
@bigshape neoprene could work - this EVA has very similar texture. Reckon PE foam sheet would work too, if you can find it in the right form. I couldn't find what I wanted on ebay and some specialists were charging too much for delivery. Also stopped by a sewing/crafts shop that had Pellon stabilizer by the yard - might be worth a look.
I think PE is something different again.
As with all foams, EVA comes in different densities and varying grades of softness. Sadly that's where my expertise ends, I don't know which one is stiff enough, but some searching around cosplayer/crafter pages should result in some recommendations.