Sounds like you're having the same/similar issues as you had with your last microphone extension lead you posted about before.
The problem with tracking down such issues is there seems to be many 'suggestions' or 'solutions' presented that are usually stabs-in-the-dark or "it worked for me" statements but never actually fixing the original problem. (Sometimes just luckily 'reducing' the effects is good enough for most.)
There's things like "RFI" (Radio Frequency Interference), to Common Mode Currents, Ground Loops, the "Pin 1 Problem", poor grounds, etc. that can all give you woes with unwanted noises on your transmitted audio. Obviously putting on a longer lead can exasperate your problems for any one of those issues.
If you like things complicated or gaining a better understanding there are many documents available such as the following :-
Good reading eh?
(Ignore the 'Bucket Treatment' - that's just a sense of humour thing.)
The 'simple' solution would be just to remove the microphone extension if it works OK without it. (Not a fix but definitely a 'solution'!)
There's bound to be other suggestions involving 'ferrites' or other such and you can fair old chase your tail with all of these let alone the 'dark magic' solutions. (It seems your battery wrapped in insulating tape may have shown up ground loop issues.)
Ultimately only you can 'fix' your problem by understanding what's causing it in the first place or as many do just trying everything in the 'book' until it's "better".
I remember an old breaker from years back that had a power/desk mic problem but everyone that turned up to try and 'help' him couldn't find a problem or failed miserably at a solution. Turns out that when he was having an evening relaxing 'rag-chew' he would kick off his slippers, place his feet on his floor mounted big linear power supply (to keep his feet warm?) whilst holding the mic by the neck resting on his gut....and the buzz would be back.
Power/Desk mic amplifier electronics don't usually help with such problems.
I hope you get sorted, wish you luck, as well as wish you well.
All the best,