Perlite or vermiculate to lighten the thermal mass would also make it less efficient - they have a lot of entrained air in them and do a good job of insulating (insulating=slowing down the process of heat transfer) meaning less transfer of heat from the exhaust before it is vented outside. You could compensate by having a longer run through the TM, but that would require more TM which would weigh more overall. There is really no way to escape the need for mass in the thermal mass - except by using costly phase-change materials like paraffin (also flammable).
Two ideas - complex and simple.
From all that I've heard what you need for good TM is direct contact with the exhaust run from a highly heat conductive material -
cob being the practical optimum - which in turn is embedded with high mass material for heat accumulation (rock, brick, stone rubble).
For a tiny house on a trailer you need something which is permanently installed - but is low weight when the home is in transport mode.
It really sounds like water could be
the answer. Very conductive, plenty of mass, very portable, easily drained out when necessary, safe and nonflammable (but provision would have to be made to safely vent steam in case of overheating). The amount you use would have to be carefully thought out - you can't let it get hotter than 100C (212F), so the total volume would have to be fairly high so that it can absorb a useful number of BTUs without transitioning to steam.
I am imagining a heat bench with a short run of conventional cob/stone TM at the hot end of the exhaust run, but with the remainder essentially a series of water containers with a relatively small amount of cob between containers and in direct contact with the exhaust duct - like a cob honeycomb. It would be quite massive in weight, but a good portion of that mass could be drained out.
But ... Perhaps you should just go simpler.
Your house, all its furnishings, and even the air inside it, is thermal mass. Assuming your tiny house is well insulated, you can just utilize the home itself as your thermal mass. When not in transport mode you can add to the home's mass by using water drums as table supports, end tables, etc. (as a bonus you have a good supply of emergency water). They'll retain the ambient indoor temperature - which in cold weather you can maintain by more frequent firings of a small RMH stove without built in TM. Yes it will be less efficient than a big thermal mass unit that recovers a large amount of the exhaust heat, but it will still be better than a conventional furnace or
wood stove.