BIC or LMM for hypothesis testing for repeated measure experiment

I am comparing two conditions of a repeated measure experiment.BIC was used as model selection criteria with a baseline model in one literature and another one used a Gaussian link function and regression but no fancy BIC.
This might be very basic question,I am not sure when to use which approach?Is there any thump rule about this?

I don’t think anyone around here would recommend using the BIC for anything. It is an estimator of the density of the data under the model after having integrated out all the parameters under restrictive assumptions. If one were interested in estimating that quantity, the the bridgesampling package provides a better estimate of it. But even in that case, you have to assume one of the two models is correct, which is not something most people who are doing (or reading about) an experiment are prepared to accept.