While I'd love to have a safe, well implemented, and user friendly way to watch replays while in queue, this sounds like a lot of work that's honestly probably quite hard to justify considering what else that time could be used for.
Hence that's not what I'm requesting here.
What I'd like is just an advanced/experimental/hidden setting that allows me to watch replays while in queue at my own risk.
Doing so is already possible if you really want: Leave your TMM party, start the replay, close the client, restart the client, rejoin the TMM party, join the TMM queue.
Similarly, the checks for the already existing watch-replay-while-in-lobby feature is a bit safer than no checks at all, as, while it tries to prevent you from loading incompatible replays sometimes, it doesn't always do so. That's very much fine though, because if you close the offending replay in time, everything runs as you'd expect. (If you don't, the game does crash though.)
So instead of a popup straight up preventing me from starting any replay, even compatible ones, while in queue, I'd suggest an advanced setting that replaces the popup with one that warns me about the potential crash, but then lets me proceed anyway.