Is the Likelihood Ratio test using cluster robust standard errors fixable by Bootstrap (or someting else)? There is a common agreement about the invalidity of using likelihood ratio tests when computing Maximum Likelihood Estimates (MLE) using clustered corrected standard errors. The main argument is that observations are no longer independent; hence, is it not a proper likelihood. The same argument goes when using weights in the estimation. However, I haven't happened to find any paper discussing said issues or offering corrected asymptotic distributions.
For instance, the official Stata's FAQ offers the following.

The “likelihood” for pweighted or clustered MLEs is not a true likelihood; i.e., it is NOT the distribution of the sample. When there is clustering, individual observations are no longer independent, and the “likelihood” does not reflect this. Where there are pweights, the “likelihood” does not fully account for the “randomness” of the weighted sampling.
The “likelihood” for pweighted or clustered MLEs is used only for the computation of the point estimates and should not be used for variance estimation using standard formulas. Thus the standard likelihood-ratio test should NOT be used after estimating pweighted or clustered MLEs. Instead of likelihood-ratio tests (the lrtest command), Wald tests (the test command) should be used.

They conclude by referring to a possible Bonferroni correction when using Wald tests. However, I am still wondering if the conventional likelihood ratio test is still ''salvable'' by, for example, adjusting the asymptotic distribution or using some sort of Bootstrap procedure. Finally, as far as I understand the problem, the issue is with the asymptotic distribution not being $\chi^{2}$.

I believe you are correct.  If we could derive the full and complete likelihood with correlated data we could construct a proper likelihood ratio test statistic.  The Wald test can be viewed as an approximation to the likelihood ratio test [1] (both are functions of the parameter estimator) and the Wald test can easily incorporate covariance terms for correlated data.
The standard issue likelihood ratio test statistic is constructed under an independence correlation structure but if the data are correlated the statistic won't follow the usual asymptotic chi-square distribution.    You could absolutely bootstrap the sampling distribution of the standard issue likelihood ratio test statistic to estimate its sampling distribution and standard error, but this is computationally more expensive than the Wald test and may not add much value.
