"Well I guess British Gas can hedge on any basis it wishes given that it has a wholesale arm that sets whatever price it wants. "
It can't set whatever price it wants - the price is determined by the international gas market.
"And it can also watch its wholesale arm edge out any smaller companies by charging whatever price it wants. "
If the gas production arm of British Gas charged a higher price than smaller gas companies could buy at in the international gas market then they just wouldn't sell any gas.
"Doesn't mean we should pay for that. "
You pay the market price for gas which is mainly determined by supply from Norway, previously Russia and Middle Eastern LNG.
"As well as pay British Gas just for being British Gas."
Don't quite understand this point.
Of the gas supplied in the UK 50% is produced in the UK by the likes of BP, Shell, Total , a third comes from Norway gas mainly Equinor, 5% used to come from Russia which is Gasprom and most likely now at zero and the rest comes from LNG mainly QGPC, Adgas, Sonotrach.