Wow there's a lot going on here... I don't know if we have any good calcs to make sense of the ridiculous number of lightyears that the cosmic pokemon from Sun and Moon fly and fight at, but the 'countless/infinite' space that's covered by Ultra Necrozma's attacks that Lunala should scale to are either stupidly high MFTL+ or is that 'likely Infinite' speed, meaning Lunala might blitz either way we look at it? (Obviously blitzes with Infinite, but the MFTL+ is unclear)
I don't know if any of the inventions Eggman has have passive activations for their abilities, or if they even do if any of them are viable, looking over the page I don't see anything standing out that its passive?
Moongiest Beam's negation properties might not be viable here, since it's in relation to damage reducing abilities. And I don't know where the dimensional BFR comes from, so I won't say that's viable either until that's better explained.
But since they're both bloodlusted, and I think there's reason to believe Lunala's faster either way we look at it, it should be able to use its teleportation/portal creation immediately to get a clear shot at Eggman himself no matter what mech he's using, and Eggman can't tank ANYTHING Lunala does from there.
I'm not sure Solgaleo's even a factor, because if Eggman's caught it, releasing it would probably take his own reaction/combat speed, which if Lunala blitzes, Eggman's dead before summoning it to match Lunala's speed.
Not totally voting, because the speed thing's unclear, and also just...This seems like a merry-go-round of stompping. Infinite Lunala seems to give Eggman no wincons, MFTL+ whoever's faster has pretty obvious one-shot options that bloodlust means they're PROBABLY going to start with, and speed equal means Eggman's hax and AP probably stomp outright. I don't think this fight has any viable way to work out the conditions without someone utterly dominating.
Edit: Reviewing SBA, I think unless it's changed by OP, we're giving Lunala infinite via that being it's highest possible scaling...So my reasoning for Infinite being a stomp is my current final answer.