So, as I understand the situation...
Your radio picks up noise all the time when at the operating position,
even:
1) with the rig powered by a battery,
2) with the circuit breakers to the house shut off, and
3) with no antenna connected.
Boy, that is puzzling...
I'd suggest at a next step making sure there are no other wires
of any sort connected to the radio, then shorting out the antenna
connector under those conditions, and see if you still have the
problem.
I've run into two cases with somewhat similar symptoms. The
first I noticed on 15m, covering 21.075 to 21.175 MHz with a
squirrely noise. Then I found it in two different segments of
10m, and started working it backwards. Turned out to be a
fundamental on something like 449.5 +/- 1 kHz, and I was
hearing the harmonics throughout the spectrum. I tried sniffing
it out using an AM radio tuned to the third harmonic, and it
appeared to be radiating from an agricultural box about 70m
away, but as I got close it swamped the IF stage of the
receiver. (There was a ham who lived even closer to it, who
didn't use the affected parts of the bands, but couldn't get
good AM radio reception for some reason...)
So if your two radios use the same IF frequency, it could be
some sort of radiation that gets picked up there, so that the
actual frequency that the radio is tuned to doesn't make any
difference.
The next case was in this house when we moved here, and
I've encountered it on other cases. I had horrible interference
that seemed to come from everywhere, but nowhere in
particular. A year or so ago, the power company made some
major upgrades, and the noise disappeared. (Well, at least
the worst of it, I still have other local sources.) Apparently
the noise was coming in on the AC power ground wire,
so shutting off the circuit breakers didn't disconnect it, but
it was still radiating throughout the house. I've seen a couple
other cases, including one where there was a loose tie wire
on the power cable about 500m away from our farm, that
apparently was arcing. It wasn't a problem when I used
a balanced antenna, but it was when I used a wire fed
against ground. And it caused a loud roar when testing
an 80m DF receiver inside the house - made it easy to
trace where the wires were running through the walls. That
one apparently got resolved when we upgraded the electric
service and got better ground rods installed.
So those are two possible mechanisms that could cause
strong radiation in the house, even with the power shut
off.
If shorting out the receiver input doesn't reduce the signal,
then it might be getting into to the IF stages directly. (The
best check would be turning down the RF gain, but only if
it affects just the RF stage and not the IF stages.)
The quick and simple Faraday cage would be to wrap the
rig in kitchen foil, but that makes it difficult to see the S-meter.
Just leaving a hole for that may still work.
But it also points out the importance of paying attention to
every wire that connects to the radio: if your battery is
connected to a charger during the test, noise could get in that
way even when the power is shut off to the house.
That suggests another test, however. What if you set up the
rig on a battery down by the antenna, connect the coax to it,
then put a small antenna on the shack end and sniff around
to see how strong the noise is? Not that it will be particularly
convenient, unless you can turn up the volume on the radio
so you can hear it at the other end of the coax, but at least
it would reduce the noise in the radio so you can tell where
it is stronger or weaker around the house. That might, at
least, resolve whether it is radiating from the power lines
themselves, or from the AC power ground wire running in
your house.