Going into a UDRP proceeding without experienced attorney(s) doesn't sound like a good plan ... but then perhaps you're get lucky like the registrant of elizabethtaylor.com did recently.
I'll throw out my layman's opinion here ...
Firstly ... UDRP the last I checked has no formal rules for discovery - some panelists (your side did ask and pay for a panel right? -having only one panelist, with some limited exceptions, is often bad for the respondent) will improvise; will research on their own various facts presented ... but one can't count on this - be prepared for the complainant's attorneys to throw in everything including the kitchen sink; attempting to challenge so many points can get involved and very often lead to slip ups that weakening the respondent's position.
With that said, you may be in luck in that the law firm that filed 'marlon brando' marks allowed one to be abandoned (serial 76417738) - they appeared to run into a problem.
And their luck with their very recent registrations may not be any better - I see USPTO responses were sent ... not sure if they were simply standard procedural responses or are for more serious issues.
Even assuming their registrations are totally shot ... a common law mark alone can spell trouble - if they can present a good case as to having one, then your side may likely be forced to challenge that point and/or claim to have a common mark as well for the name; if both sides have trademark rights, then the panel may choose to rule for the respondent ... but quite possibly they may, in such a situation, choose to not make a definative ruling either way, which could lead the way to a legal fight in court - that's where the costs can really go crazy ... have 6-figures ready to spend; in all likelyhood, the other side may not be willing to fight that hard based on my cursory reading of this thread, the marlonbrando.com website, and their TM filings ... but point is the potential of the costs esculating sky high is there - and once such a legal case is moving along, one may find bailing out to not be a viable option.
Ultimately it could come down to whether the domain is being used in bad faith - and it appears you recognize that already from what you've written ... with that said, even though the domain was allegedly given as a gift to someone, that doesn't automatically mean it's being used in good faith nor preclude them from gaining control over the domain.
Anyways, I'm rambling on - at minimum, consult with an attorney who is knowledgeable of UDRP *and* ideally has been through some UDRP fights ... best to be overprepared, then get caught short; your posts here suggests to me, perhaps wrongly, that you're unprepared for the UDRP arbitration hearing; even if the panel rules your way, arbitration isn't automatically the end - they could respond with legal action; lawsuits, etc.
Ron