Now you're thinking!don’t want the possibility to reinfect the fish in QT
Do not bring a sponge to the QT from the DT (or any other source where there may be disease/parasites).
When you set up the new, clean QT I would use a new sponge filter (or some other filter with media to house the nitrifying bacteria) and seed the clean QT with store bought bottled nitrifying bacteria. Until they grow and take effect, you may still need to control ammonia by other means.