So I decided to try just the mouthpiece for three weeks to see if it really makes a difference and the verdict is: it works!
I played about 30 minutes/day of Kopprasch with the mouthpiece only almost every day. It's not easy to play all the notes in tune but a did my best to play the etudes as accurately as possible.
When I started to play on the horn again after three weeks, the sound wasn't quite there but I had almost as much strength and endurance as if I hadn't stopped playing. The sound came back after a few days of practice.
Normally, it takes me one day of practice per day off to get back in shape. So I would have needed a couple of weeks to be in concert shape this time but with 30 minutes of mouthpiece practice almost every day, I was ready to play a decent performance after 4-5 days.
So yes, buzzing your mouthpiece on days off can help keeping your chops fit while you're away from your instrument but you need to be motivated and consistent. Also plan a few days to work on your tone and flexibility with your horn before your first gig.