Part Time

The eligibility requirements for part-time MBA programs are quite similar to the requirements seen in full time MBA programs. Just that these programs are designed for working professionals with at least two years of work experience.
Candidates will need a Bachelor's degree or equivalent in any discipline from a recognised university. They would also need GMAT or other entrance exam scores, essays, recommendation letters, a resume, and a personal statement.
However, in many colleges that offer direct admission, aspirants will not be required to submit GMAT or other entrance test scores.

Yes, you could try for distance.

GATE score is valid for 3 years. GATE score is not required for seeking admission in part-time or distance learning M.Tech.

I would suggest you go for MFM as you are already from the same background and this will also help you get a better job.

If you want to study in UK for MBA, you need to have 6.0 ILETS score and 2 years experience and the lowest fee is 20 lakhs in University of Bedforeshrien.