Reproduce the issue, but instead of turning the Coolsat off, turn off power to the LNBF via the menus. That might help isolate if it is the power to the LNBF triggering interference or the receiver itself.
If the problem doesn't go away maybe a component is failing on the Coolsat. There is a pretty common problem on the Coolsat 6000 with a capacitor in the power supply going out. If it is failing it could cause problems with anything plugged into that outlet or out the RG6. As Anole suggested maybe run a drop cord over from another outlet.
My 6000 was having a problem with the IR not working when it was plugged in near all my other gear, but IR worked fine in the next room. Of course that didn't help as the TV & RG6 were not in that room.

I replaced a couple parts and it is working now.
I wasn't sure from your statement, did the 3rd / upstairs TiVo have the same problem?
Are OTA & FTA sharing a common ground at your demarc?