Cops can already warrantlessly enter a home under the so-called community caretaking exception to the warrant clause.
And, EMTs can report anything illegal to the cops. I imagine they already do, within the limits of their own conscience of course. I imagine those reports are treated as tips, and if so, I'll bet EMT tips enjoy a higher indicia of reliability when it comes to probable cause.
And, I imagine there are still plenty of counties in America where the first guy to get to the house is a deputy because the volunteer rescue squad is on the other side of this half of the county and the deputy just happens to be closer when the call comes in.
I'd worry more about the quality of emergency medical care I would get. It might be a good idea to document his antagonism and put it on file with the town or county government.
Separately, perhaps its possible to trespass his office rather than him. For example, "John Smith is forbidden to enter onto my property when acting as an LEO." But, that would just serve to stir up more antagonism, risking worse emergency medical care.