Yan, great news about your Belansky Tatra. They are beautiful.
Would you add the wonderful photo to the database?
Rollers (Sempervivum globiferum subsp. preissianum 'from Belianske Tatras')
On to your D'Aubique.
If the leaves are not mushy, just leave them. They may just dry up and become crispy. If they do turn mushy, you can gently pull the leaves from side to side, dislodging them from the rosette. Be very gentle in how you stabilize the center of the rosette, so you don't bruise it. You may not have to take it up from the soil.
If you do have to take it from the soil, clean the mushy leaves quickly, work the soil up to make it a little fluffy, and replant.
You might try putting something over the container to keep the rain off until it dries out a bit.