Well if you don't have any problems with the amount of time then I guess that route is OK. However please note that your graduate GPA will not improve your 2.4 GPA, since graduate and undergraduate courses are viewed seperately by med schools, and in most cases, undergrad GPA is weighted more. Additionally some schools screen GPA, and tend to have a cut-off around 3.0-3.2. Regardless of your efforts in a BME program, if your GPA is <3.0, then you may be rejected right off the bat.
Doing well on the MCAT is good, but again, GPA is also important and in most cases GPA and MCAT are weighted equally, thus doing well in one may not make up for the other. Although your interest in research is commendable, you may want to sit down, think about how best to acheive your goal of getting into med school. With a 2.4 GPA, it will be tough even with the highest MCAT score, and upward trend in GPA.
I for one have a 4.0 GPA in my PhD program, and a 3.4 in biomedical engineering post-bacc work. However my undergrad GPA was a 2.65 (today its a 2.77 thanks to post-bacc). My MCAT is a 36, however even then it will be tough to get into a med school in terms of numbers...mainly due to the screening processes. On top of that, biomedical engineering courses do not count as science courses...ironically...so doing BME, although great for your interests may not add to your science GPA (but it does add to your overall GPA). To overcome these issues, I will take more undergrad classes over the course of my 3-4 year PhD program to boost my undergrad GPA to a 3.0.
For your case, I would probably (1) focus on boosting your GPA in the best way you can. Personally I would do post-bacc over a grad program if you can afford it. (2) Ask yourself why you maintained a 2.4 GPA. Some of us, myself included had personal issues that were more important than school, thus our GPA floundered, but upon resolution of these issues, we have excelled in our academics and extracurriculars. If this not the case with you, I would highly suggest rethinking your study skills. Its great you want to pursue higher learning, but you also want to do well in the program.
Getting into a grad program, especially one that is highly impacted such as BME is rather difficult even with a GPA of 3.0 and above. It may be challenging (but not impossible) to get into a BME program (or any other program) with a 2.4 GPA. So you may have to address this issue too. Lastly, wanting a strong clinical background may not be a sufficient reason for going from research to medicine. There are many PhDs that have a strong clinical background. Additionally with recent efforts to integrate medicine into biological sciences, there are even more scientists with strong clinical backgrounds (e.g., via Howard Hughes funded fellowships). I for one am a PhD student in pathology, and my thesis IS VERY clinically oriented. I diagnose patients, assess treatment, and will in the future, for my thesis, I will run an intervention study that will prescribe and change therapy (via physician intermediate) as well. So yea, I'm not trying to toot my horn or anything, but I wanted to show you that there is GREAT overlap between physicians and scientists.