I tested how He would influence the time of the cooling-down and the warming-up of the cryostat.
For testing the cooling-down, I injected 20Pa of He gas in the tank and started the cryo-pumps. Interestingly, by reaching 250K, the temperature curve started to decrease less than the curve for pure vacuum. It then approaches to a temperature of ca. 180K. By pumping off the He inside the tank, the curve decreases as usual again (first picture, red curve). It seems that the isolation around the cold finger is not very good and He started to become a heat-bridge...

Warming-up the cryostat, on the other side, becomes now a very fast thing by injecting He gas: after 50h, I reached room temperature again. This is very convenient compared to the one week (!) we needed with pure vacuum.
The temperature curve for warming-up the cryostat can be seen in the second attached picture.
(Because of the low-temperatures, when I injected some He into the tank, I could not control the final pressure properly. That is why I got 62Pa in the end and not 20Pa as I was aimed for...)