Some people are way over analyzing this. It doesn't matter if he's table hopping, or if the whole restaurant is rented out. What he is doing is simply a birthday gig. No more, no less. He's being hired to do a birthday party, not to work in a restaurant.
To simplify this more, if I did a birthday party at a park, it's not consider a park gig. It's a birthday gig. If I did a birthday part at a house, it's not a house gig, it's a birthday gig. It's quite simple.
Although I understand what you are saying, I must respectfully disagree. Not all birthday parties can/should be treated the same. For instance, the average children's birthday party will typically have about 12 kids and a few parents. On the odd occasion, there may be a few more, but generally, the performance and the audience will fit into an average living room or family room.
The props for this type of a show don't have to be extremely large in order to be seen. Indeed, the audience is only 5 or 6 feet away. You also don't tend to need a sound system (unless an effect in your show happens to require music or a voice over) since normal voice projection shouldn't be a problem in a room that size.
BUT, if you now move the party to a restaurant where the ENTIRE RESTAURANT has been booked for the event, you have a completely different performing situation. The audience may now be over 100 people, sitting at distances up to 50 feet away from you (or more). As a result, you will need much larger visible props, a sound system and perhaps even a raised platform. Your set-up and tear down times become much more extensive, and you may even need to rent a larger vehicle to get everything to and from the venue.
If you treated this simply as a "birthday show" and charged your normal birthday show rate, it really wouldn't be worth your time and effort. If you are performing in an evironment like this, it really is more akin to a small stage show.