I'm going to be the voice of dissent here. If the primary focus of this lunch is networking with Gandalf, then it is rude to exclude Mule simply because you don't like him.
Mule is not currently invited to the lunch, because frankly, Fox forgot he existed when she organized the initial invitation list, and Gandalf hasn't asked after him in several years. Ferret asked a few days later if Mule was invited but he might have been joking, and Cat strongly prefers that Mule not be invited. Fox, however, is torn. It seems unprofessional (or perhaps uncharitable) to exclude him primarily because we don't like or respect him,
If the focus of this lunch is to talk about work that Mule is not a part of, then that is different.
In a professional environment, Gandalf shouldn't have to ask the rest of the group about Mule in order for Mule to be invited. Although it's not unheard of for a contact of mine to ask about a collegue, not asking me about him/her does not equate to my contact not having a working relationship
with that person. That by itself, cannot be the basis for not inviting Mule.
I also don't think that it's up to anyone else in the group to decide if Mule is someone that Gandalf can be proud of mentoring or not. If Gandalf has mentioned that he's not proud of Mule, then, IMO, he's being extremely unprofessional.
Looking at this strictly from the perspective of a business meeting, then the criteria for inviting him is not based solely on whether or not having him there will make it enjoyable for all the other coworkers.
Now, that being said, you could justify not inviting him if the lunch is a mentoring opportunity for your particular group and is focused on a project that Mule is not working on. That way, he's not invited because the lunch is for your work group and his work group would have the opportunity to set up a separate mentoring session with Gandalf at a later date.
If, however, this meeting is simply networking for the groups at large, then it's wrong to purposely exclude Mule simply for not respecting him. (The fact that someone asked about Mule to begin with makes be believe that this may be the case).
I think that for this meeting, you need to be very clear what it's purpose is and be prepared to defend that if/when Mule finds out he's been excluded.