No, Less Wrong is probably not dead without Cox's theorem, for several reasons.
It might turn out that the way Cox's theorem is wrong is that the requirements it imposes for a minimally-reasonable belief system need strengthening, but in ways that we would regard as reasonable. In that case there would still be a theorem along the lines of "any reasonable way of structuring your beliefs is equivalent to probability theory with Bayesian updates".
Or it might turn out that there are non-probabilistic belief structures that are good, but that they can be approximated arbitrarily closely with probabilistic ones. In that case, again, the LW approach would be fine.
Or it might turn out probabilistic belief structures are best so long as the actual world isn't too crazy. (Maybe there are possible worlds where some malign entity is manipulating the evidence you get to see for particular goals, and in some such worlds probabilistic belief structures are bad somehow.) In that case, we might know that either the LW approach is fine or the world is weird in a way we don't have any good way of dealing with.
Alternatively, it might happen that Cox's theorem is wronger than that; that there are human-compatible belief structures that are, in plausible actual worlds, genuinely substantially different from probabilities-and-Bayesian-updates. Would LW be dead then? Not necessarily.
It might turn out that all we have is an existence theorem and we have no idea what those other belief structures might be. Until such time as we figure them out, probability-and-Bayes would still be the best we know how to do. (In this case I would expect at least some LessWrongers to be working excitedly on trying to figure out what other belief structures might work well.)
It might turn out that for some reason the non-probabilistic belief structures aren't interesting to us. (E.g., maybe there are exceptions that in some sense amount to giving up and saying "I dunno" to everything.) In that case, again, we might need to adjust our ideas a bit but I would expect most of them to survive.
Suppose none of those things is the case: Cox's theorem is badly, badly wrong; there are other quite different ways in which beliefs can be organized and updated, that are feasible for humans to practice and don't look at all like probabilities+Bayes, and that so far as we can see work just as well or better. That would be super-exciting news. It might require a lot of revision of ideas that have been taken for granted here. I would expect LessWrongers to be working excitedly on figuring out what things need how much revision (or discarding completely). The final result might be that LessWrong is dead, at least in the sense that the ways of thinking that have been common here all turn out to be very badly suboptimal and the right thing is to all convert to Mormonism or something. But I think a much more likely outcome in this scenario is that we find an actually-correct analogue of Cox's theorem, which tells us different things about what sorts of thinking might be reasonable, and it still involves (for instance) quantifying our degrees of belief somehow, and updating them in the light of new evidence, and applying logical reasoning, and being aware of our own fallibility. We might need to change a lot of things, but it seems pretty likely to me that the community would survive and still be recognizably Less Wrong.
Let me put it all less precisely but more pithily: Imagine some fundamental upheaval in our understanding of mathematics and/or physics. ZF set theory is inconsistent! The ultimate structure of the physical world is quite unlike the GR-and-QM muddle we're currently working with! This would be exciting but it wouldn't make bridges fall down or computers stop computing, and people interested in applying mathematics to reality would go on doing so in something like the same ways as at present. Errors in Cox's theorem are definitely no more radical than that.
I don't think it's a string of objections; it's one (reasonable) objection made at length.
The objection is that you're not really doing Solomonoff induction or anything like it unless you're considering actual programs and people saying things like "many worlds is simpler than collapse" never actually do that.
As I say, I think this is a reasonable criticism, but (in the specific context here of comparing MW to collapse) I think there's a reasonable response to it: "Collapse interpretations have to do literally all the same things that many-worlds interpretations do -- i.e., compute how the wavefunction evolves -- as well as something extra, namely identifying events as measurements, picking measurement results at random, and replacing the wavefunction with one of the eigenfunctions. No matter how you fill in the formal details, that is going to require a longer program."
(For the avoidance of doubt, the "picking measurement results at random" bit isn't reckoning the random numbers as part of the complexity cost -- as discussed elsewhere in this discussion, it seems like that cost is the same whatever interpretation you pick; it's the actual process of picking results at random. The bit of your code that calls
random(), not the random bits you get by calling it.)This is still a bit hand-wavy, and it's not impossible that it might turn out to be wrong for some subtle reason. But it does go beyond "X sure seems simpler to me than Y", and it's based on some amount of actual thinking about (admittedly hypothetical) actual programs.
(I guess there are a few other kinda-objections in there -- that Solomonoff induction is underspecified because you have to say what language your programs are written in, that someone said "Copenhagen" when they meant "collapse", and that some interpretations of QM with actual wavefunction collapse in aren't merely interpretations of the same mathematics as every other interpretation but have actual potentially observable consequences. The first is indeed an issue, but I haven't heard anyone seriously suggest that any plausible difference in language would change the order of preference between two complete physical theories, if we were actually able to codify them with enough precision; the second is a terminological nitpick, though certainly one worth picking; the third isn't really an objection at all but is again an observation worth making. But the main point of that comment is a single objection.)