Whether our bloated, privileged GPs suckling deep at the public tit want to or not, they have a vital part to play in controlling public expenditure. Their views on whether they want to be part of our immigration control system are utterly irrelevant; whilst they're paid from the public purse, they'll do as they're told.
This can be done by simple legislation making each of them personally liable for the un-recovered costs of health care given to illegal immigrants if they failed to exercise due diligence as the care path 'gateway'.
Hit them in their own purses and wallets, and few of those heady principles will stand. Most GP opponents are only principled at others' expense.