Durometers and equivalent Shore Hardness

[Moby Duck]

Sellers of "O" rings should have a Durometer tester. If you take in an unknown specification "O" ring, they can quickly test it for you to determine how hard it is.
Or alternatively contact the original manufacturer and ask them what the Shore hardness is.
The problem then becomes one of buying a suitable product for your press where the seller knows the rating of what is being sold.

Perhaps you are making this more complicated than it need to be. It sounds like a simple press that presses the soles onto shoes, how hard can that be. Sounds like it is just a big clamp with a jaw that molds to any shaped sole to clamp with even pressure. If it has a structure like styrofoam you could try closed cell foam like is used in hikers sleeping mats. Usually about 1/2" thick and they are very cheap, give one a poke with your finger and see how it compares. If not thick enough glue it together in layers with the shoe glue. Knife sheath makers use similar foam to hot press Kydex sheaths into shape which is a more complex shape than a shoe sole, and it works very well. I would consider covering whatever you make with some sort on non stick covering to stop any shoe glue contamination if this has been a past problem. Perhaps a silicone baking sheet of some sort would work.

I am pleased to read that real shoemakers still exist. Around here they have all but disappeared with the onset of our throw away society. I don't know of anywhere that they would be capable of putting a new sole on a shoe, not even the cheap stick on soles that people once put on new shoes to make them last longer, (if you could still buy them).
If the band-saw doesn't work, try an electric carving/bread knife or perhaps a hot wire cutter. I saw a rubber retailer using the electric knife years ago and it worked well with nice clean edges.