Hmmm @OhMother I’m not sure about some things you’ve mentioned.
The daughter has invited you to a family occasion so she cannot be that stuck-up.
You’ve mentioned that you are friends. Would you have met & become friends were you not neighbours ? Do you share interests, values, maybe go off to clubs or jaunts & enjoy yourself as friends?
I have the sense that you find yourself very OK & everyone else around you not quite so...
‘...very opinionated. I’m assertive...’ same difference.
‘...so I generally just laugh at the things her father tells me the daughters says.’... really ? Does he do this in jest, with a jocular twinkle ? Because if not, someone ‘asserting’ their opinion in my own home about my family by laughing at them would be out the door.
‘Her word is law in his eyes by the way. ’ likely so, he brought her up & contributed to her outlook on life. Plus she is his daughter, his flesh & blood, & trust runs deep in families.
Could it be he thinks of you as a helpful neighbour & keeps matters friendly as he is well-mannered & polite ?
I suggest you accept the invitation with good grace, turn up with a nice bunch of flowers, mingle a bit, say nice things & go before you outstay your welcome either as a friend or neighbour (meaning that you can no longer keep your trap shut or opinions to yourself) leaving the family there to enjoy themselves with their father. Please bear in mind that for now you know it is a surprise party, so the father has not expressed a wish for you to be there...so far.
If that suggestion makes you spit or compelled to assert your opinion, then don’t go at all because you just don’t get it.