Welcome to the forums.
I have the BAD 100 so I doubt there is much difference in the preamp design. I briefly looked at the manual for the 25 and the task chart is much the same. Perhaps some voltage may be different and some minor things. Anyways I was curious to see if I did the same thing with the BAD 100.
Yep, I get a similar result. It sounds more like fan noise to me on the BAD100. That sound did not change much with the gain control either. At the volume level where it becomes most apparent, I probably would not be in the room if I played the amp at that setting. It is not a silent preamp, but it is not that bad to require a noise gate either. I would assume you may be able to reduce some of that noise but doubt you can eliminate it.
If the noise is excessive though, best to try rolling in a preamp or two. Since you had it in the clean mode, that bypasses the drive circuits V1B and V3A. The usual suspects for noise in a Rectifier amp are the tone stack driver or the FX loop circuit as they make use of cathode follower circuits. More so due to the wear characteristic on the tubes. V2 is the tone stack driver and V4 is the FX loop.
Try playing through the amp with a normal volume setting with the speaker connected. Does the noise floor compete with the guitar signal? If that is the case, you may need to explore swapping a few preamp tubes to see if that changes anything.