I wouldn't shut him down if he was healthy just for the sake of getting him off the field. There's something that can be gained in experience with the reps. But it's clear he's not healthy and he's hurt that throwing hand 3 or 4 times now this season. If he's unable or limited in practice all week, why trot him out there on gameday just because he's gotten 50% or 75% better. It's not worth it. Clearly his hand is injured, he's not throwing, he should sit this week and see how he feels next week.
But do you want him risking injury running an offense that likely has no future in Winnipeg? Better to get the backups some reps, which they need, and restart with a new OC next year, I would think. QBs have been dropping like flies this year to serious injuries. One bad hit or wrong fall could be a shoulder separation, ACL tear, or concussion waiting to happen.