Indeed, press gives up more fouls than usual. Going + generally lessens fouls than if pack it in. Press used to be magic and have run it on most of my teams but gradually switched to man cause I was getting raped by fouls. But with the right players, press is still good.
FWIW, I wouldnt recommend starting 3 centers for a fastbreak/press. I see you use a lot of depth, which is fine, its just you need Speed and Athleticism in the lineups to be effective. Also, in the fastbreak offense, you need ball handling and passing from your guards.