Pretty cool idea
Live Chess Opening Choice Proposal
Basically the game would start you off at a specific move in the opening, like in the case of the Traxler Gambit, the game would start from the bxf2+ position. Players couldn't be picky and decide what move they want to start at in a specific opening, but just at the move that differentiates the opening from others, before the lines start branching off.
U can always challenge people for custom position games. But I know they may not accept your challenge. But the option proposed by you is not viable cuz pairing can be very late. Most users here play rated games (standard or variant). For example, I can easily find opponents for an unrated 10+0 game but I have to wait much longer for an unrated 5+0 game. And I have never got paired for an unrated 5+5 game.
I was thinking, maybe there could be an option where a player could select a specific opening and the color they want to play, instead of just a random new game. This would be separated from the normal live chess player pool. Let's say you feel like playing the Traxler CounterAttack as black, you select the opening name, the color you want, and it will match you against someone who wants to play that same opening but as white. Or if you wanted to play the Kings Muzio Gambit as white, it will match you with someone who wants to play that as black. This way people could get practice in specific opening lines that they want. These could either be unrated or rated, it doesn't really matter that much if it's mutual, but I guess some opening have an inherent advantage for white or black so maybe not. This would be a separate option in live chess a player could select, but in addition to matching rating/time controls, it matches people who happen to want to play the same opening!