I think the "offline option" you gave would work better as a piece than what was described with chess engine evaluations determining the piece inside, as it would be a risk factor with a bit of luck, as how the computer evaluates the position is always going to be different than how a human does, to such a degree that it's somewhat based on chance unless you are a computer. If you still want something that can go through a "metamorphosis" of sorts, I think something based on captured pieces would work better. For example, it could turn into the last piece captured by that player, or maybe any piece captured. Something more simple but still based on your advantage would be how easily you can check the opponent/how easily they can check you (ignoring checks given by that piece metamorphosing). So if the opponent can check you in one move and you can't check them in one move, then the stored piece is a pawn. If neither of you can check in one move or if both of you can check in one move, then the stored piece is a bishop/knight (you choose when it metamorphoses). If you can check in one move and they can't check you in one move, then the stored piece is a rook. The central square queen rule still applies.
Idk, just my thoughts. They probably don't mean anything much tho.
Online option
Instead of queens, both players have "sealed" metamorphs. Metamorph is an indeterminate figure whose quality is determined by the engine. The higher the advantage of the player (considered by the engine), the higher the figure sealed in the metamorph. Let's say, if white's advantage at the moment is much higher than black's, then the rook is "stored" in the metamorph. If the advantage decreases, then the metamorph deteriorates in quality inside the seal to a bishop, knight, and then a pawn. A pawn in a metamorph means that the player is much worse by advantage.
The queen in the metamorph appears only if the player managed to bring the piece to the central square e4, e5, d4, d5, regardless of the advantage.
You can open the metamorph at any time at will, but you cannot return it back. Until the showdown, he moves like a king.
All other rules are the same as in regular chess.
Offline option
The metamorph's piece automatically acquires the properties of the opponent's piece closest to it. If there are several of them, the player has the right to choose which one.