Re: MRT vs BGK
Actually in this case with periodic BC, pressure=density is obtained up to a constant (as also in the NSE systems). You can use a pressure filter/projection to get a unique solution (see https://www.openlb.net/wp-content/uploads/2011/12/olb-tr3.pdf). Maybe the MRT and BGK version uses a different force modell. You can adapt them by writing a new dynamics if you like.
Best
Mathias