The synagogue I used to belong to in another town had to make a very specific policy that families holding a bar or bat mitzvah were not allowed to restrict attendance, and that included to any reception held immediately after in the synagogue’s social hall. If you wanted a private reception, you had to hold it at another location. Want to use synagogue’s social hall, where members had to pay only a $40 clean-up fee? you had to be ready to feed all comers. To be fair, very few people who were not friends and family of the b’nei mitzvah kid’s family would stay, and anyone who wanted to host an expensive catered lunch for a small group would go to another location-- and had been for years before the policy was official. But one family one year tried to actively clear out non-invitees, including this one homeless woman who never missed a pitch-in lunch, or any other free meal at the shul, and always did come to services beforehand, and any Torah study afterwards (to this day, I have no idea whether she was actually Jewish, but we made a point of asking her to our wedding, because I don’t know how often she got a meal on a Sunday), and everyone always sort of looked after her, like making sure she was in a shelter in the winter and the hottest days of the summer, and that she had a bus pass every month. After that family pulled that stunt, they made the policy.
To be fair, there had been a couple of issues before, where families had invited so many people, that we had to set up extra chairs, like it was the High Holy Days, and we worried about having enough room for the regular attendees. But no one was essentially allowed to hold a private service there. On Saturday morning, the synagogue was open to anyone who wanted to come to Shabbat morning services. If you wanted something restricted, you were told that there were several auditoriums in town you could rent, and you could borrow a Torah from the synagogue if you were a member, and put down a deposit.
Heck, we didn’t even check for tickets on the HHDs, even though we sent them out. We always had tons of crashers.
But the point is, without the synagogue board instituting that policy, it was ambiguous. The hosts might actually be within their rights to exclude someone. The rabbi’s presence means nothing, because the rabbi has no control over the property. He or she is not like a church rector. The synagogue board controls the property, and ad hoc decisions are usually up to the president, who is usually present at b’nei mitzvah, but may not be.
That’s in the US, though. YMMV.