If you want a job at PhD level, biomedical. If you want to get a basic BSc degree and work asap, ME.
Since it is clear you want to work in industry and help build the equipment, I would not recommend a path as as a biological or chemical scientist. When it comes to biosensors or special fluids/colloids, a scientist will have to do most of the theoretical/R&D world there.
Being in charge of production or quality control or solving engineering problems once a basic design exists, there is where the engineer comes in. Though, a PhD in biomedical engineering can be involved from the start. Not clear if you want an R&D job or something else. R&D jobs are generally harder to get.
I think a mechanical engineer fresh out of school has basically no affinity with the R&D side of biomedical. You can check requirements asked in vacancies in your area. Always possible to get some job at most companies with a BSc in ME. Though with a master in ME and experience in the field, it can be just as good as a MSc in biomedical.
Remember that for these engineering fields often a lot of knowledge exists only inside said company, so job experience is very important and may in the end trump degrees. But of course you can't just choose to get a BSc in ME and get 15 years of experience.
When you arrive in a company, the biggest expert there may be just that type of person. Doesn't mean that right now with degree inflation you will have the opportunity to follow the same track.