If you're asserting a right to X, where X violates someone else's property rights, X is not actually a right.
Usually when I run into the "conflicting rights" claim -- or attempts to get around it with the "personally bubble" fantasy, or the "it violates my rights to put condition X om me getting something I waaaaaaaaaant, because it's something I waaaaaaaaaaant" claim, it's about guns.
At the moment, it's mostly about vaccines, though. It goes something like this:
"My medical situation is my business and nobody else's. Nobody has a right to tell me that I can't enter his store unless I'm vaccinated -- that would violate my right to medical privacy."
No. It. Wouldn't.
The rightful owner of property has the right to set conditions for the use of that property, full stop.
If the condition is "you may not carry a gun on the property," or "you may only enter the property if vaccinated," or "you may only use the property while naked, mounted on a pogo stick, and continuously singing The Star-Spangled Banner," that violates no rights of yours whatsoever.
Here are your choices:
- Refuse to comply with the conditions and don't use the property;
- Comply with the conditions so as to be able to use the property;
- Use the property without complying with the conditions, thereby making yourself a force-initiating trespasser who is stealing the use of property that's not yours.
The property owner may be wrong, or even unreasonable. That doesn't change the fact that it's his property, not yours. His conditions regarding the use of his property do not and cannot violate your rights.