1. What's the strongest darkness/shadow interaction a Beholder can resist?
2. I'd question how well Enervation (Draining energy), Death Manipulation & Disintegration would work on a Ghost-type, especially Mimikyu's True Form, supposing it's akin to some kind of undead, but that might be a bit speculatory. Mimikyu's profile also lists Non-Corporeality.
3. Moves that might be relevant:
Astonish: Attacks via a startling shout. May flinch. Range, but otherwise lol Fear resist.
Wood Hammer: Attack that seems most likely to use the tail stick. Recoil. Other candidates are: Scratch, Feint Attack, Slash & Play Rough.
Hone Claws might help with hitting somehow, but there likely isn't an opportunity to use it.
Pain Split is beneficial because of Beholder's higher stamina, but stamina doesn't help against most effects. Description:
"Misdreavus slams its body into the opponent. Then, a beam of energy comes out of the healthier Pokémon, transferring its health to the weaker Pokémon."
Curse: As a Ghost-type, using this knocks out half of Mimikyu's Stamina to place a Curse on the opponent, which gradually depletes their health. In game mechanics, it's 25% of max per turn.
Bulbapedia's description of how it appears in the manga is :
"Gastly's opponent starts suffering from an intense pain and becomes unable to move."
I couldn't find any listed appearances of it in other media.
Nightmare. Same damage rate as Curse, but only works while the victim is asleep. Descriptions:
"A move that makes a sleeping target have bad dreams. The victim will steadily lose HP." (Stadium 2)
"A sleeping foe is shown/sees a nightmare that inflicts some damage every turn." (FR/LG to US/UM.)
"Haunter waves its arms and black energy comes out of its body and surrounds the sleeping opponent, giving them nightmares." (Manga)
Of course, Beholder probably isn't going to sleep unless Mimikyu Mimics/Copies its Sleep Ray.
Baby-Doll Eyes: Priority physical AP reduction. Probably Empathic Manipulation based. Charm is more severe physical AP reduction, but without the priority. Probably irrelevant because magic.
Grudge. I'm not sure how we treat this move. In game mechanics, if the user faints immediately after using it, it depletes all the PP of the move that did so. However we translate this, it's practically sitting duck suicide for a minor inconvenience. Easily Mimikyu's worst choice, but I dunno how likely it is to use this in-character.
Destiny Bond: If Mimikyu faints, the foe also faints. I'd presume similar is true for death. Descriptions:
"The foe faints if the user does."
"After using this move, if the user faints, the Pokémon that landed the knockout hit also faints. Its chance of failing rises if it is used in succession."
"Misdreavus's body becomes outlined in purple. After fainting, two swirling purple lights come out of Misdreavus's eyes to the opponent that knocked it out. The opponent becomes surrounded by a purple aura and instantly faints." (Anime)
"Destiny Bond, a Ghost-type move. When a Pokémon faints after using this move, the opponent that landed the knockout blow also faints." (Anime Rotomdex)
"After fainting, the opponent faints along with Misdreavus." (Manga)
"When Qwilfish is about to be attacked, it disappears, then reappears behind the opponent. Qwilfish then rams its body into the back of the opponent." (Manga)
Note that some of these are a bit suspect. The animations between games, anime & how they're portrayed in the manga all differ.
Anyway, it would seem Mimikyu's best options are:
A. Rush in with Disguise as-of-yet, unbusted and smash with Wood Hammer or whatever, supposing it has the AP Advantage.
B. Curse immediately, then Pain Split/dodge to keep Stamina up while Beholder is drained.
C. Try to Mimic/Copycat Beholder's own rays, likely also using Destiny Bond in between those, as Destiny Bond, at least by game mechanics, is less likely to succeed if used in succession.