In your shoes... fuck no, do NOT put this money in the communal pot.
Your ESA is your income.
PIP is not considered income as far as tax, other benefits, etc.
It is to help you with the EXTRA costs that come with being disabled, to pay for therapy, extra food, extra heating, equipment etc etc.
Some of us have to spend it on day to day normal bills but that is not what it is actually for.
Absolutely do NOT give control of this money or the backpay (unless he paid for some equipment for you I guess) to a man who is not helping you meet your needs, but is instead trying to disable you further by denying your disability and your needs.
Go back to your GP and ask for a referral to Wheelchair Services.
Tell them (GP and WCS) that you need to access education to expand your skills for potential self employment/employment later on. Acessing education and employment is a key criteria for power wheelchair provision across most of the country.
You do not have to be accessing those things NOW... you can't, but that is what you want to do in the future, you're being stopped by your lack of independence.
I have a powerchair on this basis - supplied by NHS.
Even if you do not currently need a powerchair as you have supplied your own, they can still supply the correct cushions and accessories, but they may well provide you with one at which point you can keep the other as a spare or sell it (if you have room, id have a spare, but it is a luxury many cannot afford!)