This has been a problem for a while, reported numerous times. The problem is bound to worsen the more FS releases additional cosmetics.
I once took the chance to ask hedge directly about this in October on the official discord and he answered thusly:
We don’t have plans on increasing the cap, but we do have plans to shift some items to an alternative storage solution, as you say, akin to illusions. illusions used to take up inventory space but this was changed so in the future expect other items to follow suit, in particular items that cannot be customized (hats or hero skins etc).
I have the same issue, I don’t really consider myself a hoarder even though I do have an assortment of veteran items per character, but that’s less than 1 page per char. With this new update I can no longer open chests, therefore no longer forge / upgrade veteran for the Warrior Priest.
Surely there can be some adjustment made for the veteran players who like to have a set of weapons for different builds on inventory.