Sry for late response. Anyway, here's how I think this would go:
I don't exactly recall Arjuna Alter's personality, but iirc he's a lethargic bot for anything that isn't Karna, so he's likely not to take Lucilius seriously. What he does is reset his Lostbelt (which iirc isn't as big of a feat as doing the same with an entire universe according to Da Vinci) by destroying it and remaking it with his Hindu pantheon authorities. On the other hand, what Lucilius is trying to accomplish, should be capable of doing, and is being actively prevented from enacting is the complete destruction of the entire world (his verse) by turning everything into nothing forever (no plans to survive or leave anything intact at all) to spite the Gods and have peace.
Another funny thing: Shiva, with all his Authorities over Destruction, Rebirth, and Salvation is also an entity and Primal Beast in the Granblue verse. However, in comparison to Lucilius he is just nothing, as Lucilius' own powers come from somewhere higher and are more 'potent' so to speak.
One thing to establish before the fight is the actual fighting conditions and the battlefield. IIRC neither of them need oxygen and can survive in the vacuum of space, so that's not a concern. But what should be noted is Arjuna's Vritra invulnerability authority, which makes Arjuna invincible in the following conditions - can not be harmed by wood, metal, or stone, dry objects nor wet objects, and cannot be attacked during the day nor the night.
Now for the fight:
Researcher and Reborn forms are just too weak in comparison to Lostbelt King Arjuna (you could make an argument vs regular Servant Arjuna, but that's a different topic), so they get pwned in a straight fight. However, there's a caveat - Arjuna may not be able to put him down for good, as Lucilius has deathless immortality even in Researcher form thanks to being an Astral. And while Arjuna has his Shiva Authority, granting him the power to destroy all of creation with Mahapralaya, this seems to expressly be a type of existence erasure rather than any shenanigans with the concept of destruction itself. And wouldn't you know it - Reborn Lucilius has Primarch Physiology, granting him resistance to existence erasure, which Arjuna's may not be powerful enough to overcome. Perhaps Arjuna could incap Lucilius but not kill?
Though it should also be noted that if Arjuna WAS somehow capable of successfully inflicting the concept of destruction or rebirth onto Lucilius, it may (according to Granblue lore on Primals and Astrals) cause a paradox and turn Lucilius into a Precipian Entity and power him up instead of actually killing him, so there's that...
Twelve-winged Lucilius is where things get real. He always opens with Paradise Lost to instantly end the fight if possible, and here's where the events branch:
1) Vritra's invuln condition is active, upon which Arjuna just tanks this version of Paradise Lost. Because of this, he may keep underestimating Lucilius while trying to erase him with normal attacks without bothering with Mahapralaya, and would find himself getting assblasted by The End after Lucilius' Countdown to the End ticks down to zero - whoops.
2) Vritra's invuln ISN'T active, and Arjuna gets blasted in the face with Paradise Lost, which he may or may not survive. If the initial volley doesn't erase Arjuna (be it defense from something like Surya's Authority which he isn't likely to activate anyway or his guts skill), the Golden Fruit that could get applied afterwards just might do him in. At this point, Arjuna would likely be forced to take Lucilius seriously and instantly try to charge up and fire off Mahapralaya, which even if successful would get hard stopped by Wings of the Word.
From 2 should follow the most 'animated' scenario for their fight, where they both would try to shoot everything they've got to kill eachother, but imo Arjuna would struggle a lot with figuring out how Wings of the Word works, before getting stopped by Seven Trumpets and then The Ended. Now, Arjuna isn't a wimp and has his own immortality, specifically this one - Gods can only die after all who remember them are gone, but with The End this shouldn't be a problem at all, though it could give Lucilius a headache if none of his other abilities could bypass this, which would drag out the fight.
Some additional notes on Lucilius vs Vritra's invulnerability: It's entirely possible Lucilius' power over Chaos could just kill Arjuna through this Authority with it's logic-ignoring properties, not to mention Lucilius could theoretically just drag Arjuna into Point Zero akin to a reality marble, making Vritra's Authority conditions irrelevant, removing home turf advantage, among other things, etc.
As for Zero Lucilius, or if he's allowed to transform? Arjuna just gets bodied - instantly.
TL;DR:
Arjuna may potentially incap Researcher and Reborn forms but not kill;
Twelve-winged Lucilius would at worst mid-diff full power Arjuna;
Zero Lucilius negs.
