Fellowships or residency programs may be difficult to arrange as they generally are geared toward a current or recently graduated student as you concluded as well. There are some Certificate programs in Healthcare which may supplement your MBA and be less lengthy and expensive than another degree, while adding some relevant knowledge. You may also need to look at volunteering in a healthcare organization to gain some experience.
You do have skills in finance and real estate that could be transferable to those specific areas of the healthcare field or to consulting and your MBA certainly provides some solid business know how, but with that said, the competition for jobs against those having healthcare experience will be a challenge. No simple asnwer to this question, but refer to the Career Services portion of the ACHE website for additional resources that may be helpful. Mike Broscio ACHE Career Resource Center Director