This being non-terminal scares me a little into thinking it should cost $5, but it would feel weak there. The presence of any draw card could make building with this too fast...
And tracking the trashes you've done should be OK?
The trash is an unordered multiset. As soon a card is added to the trash, you lose track of it. The game mechanics have no way to find out, which instance of card you have trashed and how much it costed when you trashed it. This also has issues with Rogue, Lurker, Graverobber and Possession.
Huh? There is no mechanical issue at all. This card works precisely like Smugglers, i.e. you have to remember what you gained/trashed. And it is actually on a practical level simpler to memorize/handle than Smuggler as you can put all the cards you have trashed this turn onto a slightly different place than the trash (and after your Night phase, you move them onto the trash heap).
So, what about the following case? You bought a card from the Black Market, trashed it, gained it back with Lurker and then masqueraded it to another player and then summon Modeller. There is no pile for the card from the Black Market and it is hard to keep track of the card. Maybe I am thinking too complicated or I have another understanding of the lose-track rule. Modeller explicitly requires to keep track of all card trashed in your turn, which would be a new mechanic. This is not bad, but it may have the mentioned issues.
You are misunderstanding /misusing the lose track rule. The lose track rule can only prevent a card from being moved. It cannot prevent anything else, such as finding out information about a card. You are correct that you may be able to physically find the card that you trashed, but that’s ok, there is no need to physically find it. You just have to have remembered what card you trashed, and how much it costs. The game can require players to remember things, it's ok if you can’t physically locate the card. You can choose it without seeing or finding it.
However, your edge case does make me think of a different edge case... you trash a Band of Misfits, then player Lurker to gain it back. Then you draw some cards, making you shuffle. Then play a Band of Misfits as something. Play Modeller, choosing Band of Misfits. You now need to know what it cost. If it’s still somewhere in your deck, it cost $5. If it’s the one you played, it costs less.
So yeah, “name a card” could be a fix to that, because if you name Band of Misfits, that has a specific fixed cost in Dominion of $5; and that remains true even if there are no visible Band of Misfit cards to be found to check the price.