I forgot to send a report about the service I ended up using. So I tried this Eelway's mobile luggage storage and it was quite simple & reassuring.
The concierge (as they are called) sent us a text an hour before arriving and collected our luggage at our rental. He sealed each bag and asked me to sign on his phone to confirm the seals numbers.
5 hours later at the airport, I met him at the rendez-vous point I received by email and by text. He was easy to spot with his branded yellow tee-shirt.
Another signature and done.
Very happy with this. I was a bit worried at the beginning but we had a chat for about 10 minutes about their start-up before leaving.