You said it already; CCNY-Grove Engineering admissions is quite competitive. That being the case, you are confronted with options that are cumbersome;1) pay more to attend another NYC-area engineering school (if you win admission; 2) consider affordable engineering schools in other parts of the country.
Additional Community College credits may not be useful to you. Your strong grades in your foundational mathematics and physics courses should/may convince some engineering schools that you can handle their program. Engineering departments at most state universities would cost you at least $30,000 (probably more). There are some relatively low-cost state schools where you have a reasonable chance of admission;
New Mexico School of Mining and Technology
South Dakota School of Mines & Technology
Louisiana Tech University
U of Wyoming
Boise State University
U of Idaho
South Dakota State University
The joint Florida A&M University-Florida State University School of Engineering is a possibility, as is the potential to gain state residency in Florida if you relocate there 12 MONTHS prior to matriculating at FSU.
Also, while SUNY New Paltz does not have an Mechanical Engineering department, it recently established an Electrical Engineering major. For many employers, ME and EE overlap, therefore a B.S.E.E. could be a solid base for a career with a large industrial firm like an automaker or computer technology firm (think United Technologies, Intel, Google, etc.). Of course, for any good first job at a premier corporation, excellent grades are a must. Don’t forget about SUNY Maritime College either. The Engineering at SUNY Maritime is tough, as it is at most engineering schools. Keep in mind that students are not required to enroll in the student military corps at SUNY Maritime. There are plenty of civilian students at the college.
Good luck.