I love the idea of building up your "deck" and the longer-term strategic decisions involved. Some food for thought:

Perhaps you could progress by making upgrades to your shop that persist throughout your run. You would start with a very basic shop and could have upgrades like: Unlocking modifiers, discounts, improving symbols/modifiers, locking shop entries, increasing the odds of certain modifiers appearing, increasing shop size, etc. You could also add persistent "artifacts" that give you bonuses to your die faces, rolls or abilities such as: Adjacency bonuses, modifiers for nth die face, debuff barrier, bonuses when rolling certain combinations of symbols, increased ability charges, ability gem discount when low health, etc. The score earned in a scenario could have a minor effect on the amount or quality of upgrades you can draft. 

I think this would retain the core mechanics you mentioned while still allowing for incremental progress. With some careful balancing in this progression you might be able to avoid making drastic changes to the scenarios

That gives me an idea. At the end of a scenario, you get shown all your current dice, and a set of "new" dice (the base versions for your class). For each die you have unlocked, you can grab a pair of adjacent faces and replace 2 faces on your new die with them. Effectively a free couple of upgrades based on your level progress.

I'd imagine if you were to keep your dice and stats between levels, it would need a different kind of progression altogether, and enemies would need to scale more to still make it challenging when you would keep pretty good dice and would keep improving them through each level.

An idea could be to not get any additional dice during each level like now, but instead either auto-receive more dice between certain levels, and/or having upgrade perks you could select between when advancing to a new level. Such perks might be adding one active dice, get a smattering of extra symbols on your current faces, get an upgrade for a certain kind of symbol, add debuff resistance, get a permanent free shield each round, get a bit of regeneration each round or something like that.