Boston University has a good LEAP program - check them out. I spoke with a few students that went through LEAP successfully and are now PhD route in Mechanical.
If possible try and gain either research or industry experience in engineering prior to applying. Mid to top ranked programs will want more than just coursework and good grades/GRE. Also, remember to apply to several safety schools, a few mid-tier, and a few 'reach' schools. As a guideline, mid to top ranked Mechanical programs will want 3.5+ GPA (preferably 3.7+), and 80th+ percentile (preferably 90th+) on the GRE quant, minimum. It would also be helpful to find what research areas within engineering are of interest, and maybe spin your personal statement and relate it to your psychology background.
In my experience, I think BME is a bit more competitive than Mechanical for PhD. Overall it looks like BME programs have more applicants (coming from areas in biology, chemistry, engineering, etc.), thus schools simply filter "possible admit" by GRE, GPA, publications, etc. Moreover, engineering programs are quite interdisciplinary in nature, so you would likely be able to take coursework both in Mechanical and Biomedical/Bioengineering, if it fits your research interests.