Oh boy, knew this one was somehow coming. Now it's time for a...controversial opinion.
I'm torn a couple of ways, depending on how you interpret a few things, also depending on whether this is the anime or the manga. The biggest problem for Vegito is going to be the time limit.
Manga:Assuming we take Shin at his word--then SSB Vegito > Beerus. That's it. Game Over.
But, this is only assuming we take Shin at his word. First of all, he's not even 100% sure that Vegito is stronger. He's simply speculating.
Even if we disregard that and assume that he wasn't speculating and actually meant it, Shin has not exactly been the most reliable judge of PLs in the past. If all the Kais assembled there together had made a combined statement to that same effect, or if Whis had made a nod to this power being greater than Beerus, I could find it more credible. As it stands Shin is IMO an unreliable narrator. He's an inexperienced Kai who couldn't even grasp the extent of Goku's SS2 power back when they were attacking Babadi's ship, despite his ability to casually read minds.
We also have some theoreticals thrown in the prompt, like Counter-HAKAIs and Mastered SSB. I don't see either of these happening. Why? Because Vegito, despite actually
knowing about the time limit the second time around simply chose not to use Mastered SSB. I have no idea why he'd choose not to do this. The whole point of Mastered SSB is about containing the energy of SSB within instead of letting it leak out on top of getting a major power boost. Had he used Mastered SSB against Zamasu he quite literally would've remained fused longer and decimate Zamasu that much faster. I can only surmise Vegito had more than his fair share of Vegeta's overconfidence combined with Goku's trait of letting his guard down or something, which is quite an ill-fated combination. Mastered SSB also requires immense concentration to maintain. Any slip whatsoever from Vegito means more energy lost, and less time spent in a fused state.
As for the HAKAI, I don't see this as an effective Counter-HAKAI strategy from Vegito's end, for a couple of reasons. Goku's use of the technique quite simply isn't as good as Beerus when he uses it. That's not a knock on Goku, it's just what can be expected from someone copying the technique of a GoD and then the actual GoD using it. Goku had to focus
all of his SSB energy into the HAKAI and even then it wasn't nearly as efficient as Beerus, who doesn't even look winded when he does it.
When Zamasu was HAKAI'd by Beerus in the manga it wasn't like the anime where Zamasu slowly screamed in terror as every molecule of his being was painfully erased while epic background music was playing. In the manga it was
instant. One panel he was there, the next he was getting HAKAI'd, the next he was gone.
When Goku tried pulling the same trick it was not instant. Zamasu had the time to think of a human shield strategy, open a portal, portal-grab Mai, then use her as a shield before he'd even gotten halfway HAKAI'd. Goku even had time to speak a few sentences!
This next bit is speculation drawn from the anime, but you know how Golden Frieza was able to resist that tiny bit of Destruction Energy? What if Vegito puts all or even just a good portion of his energy into a HAKAI, and then Beerus just
resists it? He'd have more claim to resisting Destruction Energy than any mortal after all. Now Vegito has wasted a large amount of his power and his very limited time while Beerus can just HAKAI him in return. Even if they get into some kind of HAKAI beam struggle, Goku's more inefficient use of the technique would mean Vegito is inherently at a disadvantage for using a more flawed HAKAI against the perfected one Beerus uses. Such a beam struggle would also raise the stakes even higher for Vegito, as the more energy he puts in to either counter or try to overpower Beerus means he's getting ever closer to de-fusing. Double this if he's also concentrating on maintaining Mastered SSB at the same time.
Finally we have Merged Zamasu. Yes, Merged Zamasu. Good ol' MZ. He's important because he shows how fragile of a being Vegito is. Not physically, but because of Vegito's time limit. Think about it for a second. Go back and read the chapter where Vegito fought MZ if need be. Now ask yourself...what, if anything, did Vegito do that could be considered exhausting? The anime had him do big, flashy stuff like a Final Kamehameha. The manga did not do this--Vegito tried but
never actually even fired the thing. The manga literally just had Vegito punch him around a bit. The most draining thing you could say he did was make that energy blade of his once and a couple of ki blasts, unless you want to count just
charging the Final Kamehameha. Then Vegito de-fused. Furthermore, he de-fused before Zamasu started de-fusing despite Zamasu's head start on him and ripping off Janemba's portal punches and throwing katchin blocks and ki blasts and stuff. Finally, the excuse can't be used that he was tired before the battle began since he snacked on a senzu before be began his Zamasu beatdown, meaning Vegito was at 100% stamina from the word "go".
I bring this up because if HAKAI was so draining for Goku, what's to say that Vegito won't de-fuse just from charging the thing like he de-fused just from charging up a Final Kamehameha, even if he is stronger than Beerus? Because even IF Vegito were to be stronger than Beerus, it won't mean much if he just de-fuses because he's so powerful and then an angry Beerus just HAKAIs Goku and Vegeta for their attempted murder of him. Vegito would lose by default since he just can't last that long as a fused being.
"But Axalon! What if Vegito just goes all out from the beginning? If he IS stronger, he can just win in those few moments, right?"
Well I'm glad you asked. See, Vegito
also declared he would show Zamasu no mercy and that there would be no time for Zamasu to regenerate.
Meaning his battle with MZ is a perfect example of Vegito just trying to go for the kill,
...as opposed to Buuhan whom he was deliberately toying with so he could rescue Gohan, Piccolo and the others.
In short, I don't see it happening. Goku and Vegeta would have a much better chance of just getting on Beerus' level on their own and not worry about de-fusing in 5 minutes or less.
Anime:I'll avoid Ultra Instinct for now, simply because I think right now would be a bit premature for using it and we honestly still don't know much about it. Waiting until Jiren is defeated would be better, IMO. So let's look at anime SSB Vegito fight MZ.
Anime Vegito actually fares a bit better than Manga Vegito in the stamina department. Anime Vegito actually is able to FIRE a Final Kamehameha instead of charging it and de-fusing, on top of using things like Instant Transmission, his blade, trading punches with MZ and blocking a large ki blast from MZ with one hand without much effort. He does fare a little worse in the fight, a TINY bit, since Manga Vegito wasn't even scratched by MZ while Anime Vegito is actually scuffed up a bit. Nothing too bad for Vegito unless you also consider--in relation to Beerus, where is Anime MZ? Is MZ stronger than Beerus? Weaker? Because if he's weaker then he's successfully tagging Vegito every now and then, even if he's losing the fight overall.
As for everything else, I figure the same concepts apply from the manga as they would to the anime, just with minor differences. Without a Mastered SSB, Vegito would have to try and boost his power through use of the Kaioken technique. Unfortunately for Vegito, the Kaioken also has a draining effect meaning it would--you guessed it--shorten his fusion time.
In short, even if Vegito does actually surpass Beerus--the levels of power he's wielding mean Vegito quite frankly isn't going to last very long at all before he just de-fuses. So even if Beerus is weaker, he really just needs to survive for a few minutes and then victory is guaranteed over the individually weaker Goku and Vegeta.
That's my two cents.
*Braces for counterarguments*