I am runnig a LCGA (negative binomial with zero inflation) for 5 waves of data. The output for the favourite 6 class solution shows that exact the loglikelihood value is not exactly replicated (10 best value see below; STARTS option was finally set to 500 50). but interestingly the usual warning („THE BEST LOGLIKELIHOOD VALUE WAS NOT REPLICATED“) does not appear. Using the associated seed value for further analyses lead to calculational problems. So i am thinking to accept this solution. Now, could it be that the loglikelihood values are too close together to suggest that the solution is not trustworthy and reached a local maximum? I suggest there is no rule of thumb to decide how many replications are desirable.
It sounds like you are using numerical integration. With numerical integration, the check of the closeness of loglikelihoods is less strict. Please send your full output and license number to firstname.lastname@example.org.
Mplus uses alpha as described in the Hilbe book. See the Topic 2 course handout.