It does sound likely to be a voltage drop issue. And the fact that it happens with either the fuel pump or radio transmission actually suggests that neither of those is likely the problem in itself, but rather that just turning on a heavy electrical load causes a dropout.
Can you get it to happen by turning on other heavy loads, like landing lights, or pitot heat?
It is possible that an in-line noise filter may have a high enough series resistance to exacerbate a low voltage situation. But no reason to jump to that conclusion quite yet. Could be any of a thousand other things going on.
So anyhow, I'd suggest that you take a more systematic approach to investigate the problem. As a good starting point, see if you can reproduce the problem in a simple and convenient environment, i.e. on the ground with the engine not running. Then, under those conditions, you can continue to characterize and isolate the problem efficiently and safely.
And by the way, running on battery power only (i.e. without the alternator supplying power), any voltage drop issues should be even more apparent, if that is in fact the nature of the problem.