I don’t think delaying the source track will help here - you’ll probably end up in a vicious circle chasing elimination of time-alignment and phasing/ghosting issues, further on in the production process. Bleed of this nature really needs, as you say, engineering out/controlling at the recording stage - if that’s the aim of the session…
Firstly, maybe you could investigate applying some ‘ambience reduction’ techniques from each of the source mics/material. There are plugins designed to help with that specifically. This may help to cut down on all the combined ‘mush’ the room may be creating, causing lack of focus/instrument definition. Failing that, you could also achieve similar by giving individual tracks very careful EQ adjustment to reduce/control the most strong neighbouring instrument/ambience bleed, without ‘ruining’ the source too much. Even basic hi/lo cuts can have quite dramatic/useful results.
Lastly, depending on your aims, be careful where the times the performer isn’t actually playing, you don’t eliminate bleed from their mic so much, as to create sound ‘holes’ of zero ambience in the overall soundscape. It might be desired, but I’ve found can also be unsettling for the listening experience.
Anyway, a few thoughts - good luck.!