Thanks for the reply! Much appreciated.
I am working on another version of this model by directly defining a multivariate probit - not indirectly via regression. With this version I am not having identifiability problems. This could be because the diagonals of the variance-covariance matrix are fixed to 1 (i.e. correlation matrix), or it could also be because, for the case of 2 tests, it actually requires less parameters to estimate (1 correlation in diseased group, 1 in nondiseased, for each study, whereas the model in this thread needs 2 in each group).
You can see my thread here.