I have a very similar height at my local refuse centre. However if I speak to the staff they open the barriers to let you in, as said it’s to stop commercial vehicles.
Unfortunately that won’t help with car parks. I get round this by not taking my bigger vehicles anywhere near a city. This is mainly because they are my nicer vehicles and cities are normally full of not so nice people who like to damage them, steal them, steal from them, or they can’t drive properly or don’t give much concern about the sh*t box they drive and don’t tend to give to much respect to what you drive in turn !!