I'm applying for a restaurant that also is my go to breakfast spot. The interview is early in the morning and I was thinking of knocking out 2 birds with one stone and ordering a bite to eat after the interview is over but I don't know if that could be seen as unprofessional or make me look bad in some way.
If you've been going there for a while then it seems perfectly normal to eat there afterwards - as long as it doesn't look like you're spending an excessive amount of money or leaving an unusually large tip as a slightly indirect kind of "bribe".
But it's worth considering that if the interview goes badly, it might be a bit awkward...
I would say that is not problem. The main reason is that you wrote that "that also is my go to breakfast spot". Most workers at that restaurant already know that you are a regular customer. They like the fact that you enjoy their food, and appreciate your interest in their restaurant. Maybe, even the manager already knows that you like the food/restaurant.
I don't see why you have to go somewhere else for breakfast on the interview day while you already eat breakfast regularly at that restaurant.
It is just another day, and another breakfast at your favorite spot (regardless of the outcome of the interview). If you don't get a job at that restaurant, you will eventually get another job somewhere else. It's no big deal either way.
Another way to think of this is: Would that restaurant rather lose 1 customer on the day of his interview ? While losing 1 order is not a big deal for a restaurant, they would rather not losing any customer any day.