You won’t be able to modify and re-sign tokens, but you could request new tokens with the new claim - To do this you may want to look into triggering silent authentication as a means to request new tokens for a user that has an existing session. Silent auth will run any Actions you have configured to add the additional/new claim. The only pitfall I see here is rate limiting which could happen rather quickly if many users are switching channels frequently.
You could use a backend to interact with the Management API and update user or app metadata with a channel_id and add that as a custom claim using an Action. Basically you would update the metadata for a user with the new channel_id when necessary, and then trigger silent auth - The new metatadata will be pulled into the resulting tokens as a custom claim.