Biology, biomedical engineering and public health are all in different schools at Rutgers. So OP would have to apply to be admitted as an internal transfer student to the destination school. Each school has different transfer requirements that are outlined on their website.
I only have second hand knowledge of all this, but I’ve heard transferring into RBS is competitive and therefore not guaranteed. SAS, SOE and SPPP are supposedly easier. But engineering requires a lot of classes, so transferring might result in an additional year at Rutgers.