I would urge you to talk with admissions people in specific ME departments to see how much flexibility there is in applying for the MS without an undergraduate engineering/physics degree. It may be that you could identify key engineering undergraduate courses to take before applying that would not only show your seriousness for working in graduate-level engineering, but would also prepare you. Other key elements of an application would be your GRE's, your statement of purpose (that should highlight what has lead to your interest in engineering, that describes how you have been preparing yourself, and how your work experience finance/accounting education are assets), and letters of recommendation.
That's great that you are interested in mechanical engineering. I'm surprised to hear that some schools have discouraged you from pursuing a 2nd bachelor's. I would check with other schools. I know that my school would welcome you! It is possible to get a MS in mechanical engineering with your background, depending on the school. But you might need to take about 2 years of preparatory classes (perhaps some additional math and science plus sophomore/junior level ME courses like statics, dynamics, thermodynamics, fluids, heat transfer, design, vibrations) before starting on the MS classes. Good luck.