1. Use signed downgrader
2. Install 6.20 TN-D
3. Update to 6.35 if you don't like 6.20 TN-D  but I think this will not happen

P.S.
If you need I can upload all the necessary files from russian forum

EDIT:
Hmm...
There is an opportunity to downgrade to 5.03 OFW if the motherboard model of your PSP is ТА-092v2
and install 5.03 Prometheus-4!!
There is another way...

1. Try VHS Menu -> USB Device to UMD Disc and dump ISO from PC, but the image is likely to be cuted
2. Downgrade to 6.20...

You mean 6.20 TN-D?

Is there no way to get the usual tools, ISO Tool or PSP Filer, to work uner 6.35 PRO?

I made the move from a PSP-1001 with 5.50 GEN-D Prometheus-4 to a PSP-3001 with 6.35 PRO-B3 (did so because the brighter screen is easier on my eyes).

Now I’m running into the problem I cannot dump UMDs with 6.35 PRO-B3. I use ISO Tool v1.970, but with this CFW it doesn’t seem to access the UMD (I see stars where the name would be with 0 byte file size). I did a test run with PSP Filer and I can’t access the UMD at all.

I’m a little (read: a lot) out of the loop with the new firmware’s, so I’m guessing I’m missing a plugin or step somewhere. Hoping someone can provide me some insight so I can get back to dumping UMDs for the list (I have a few new ones here on my desk)

Regards.