With regard to getting a promotion in KVS from your current teaching post as PRT to TGT, the recruitment rules of KVS are very clear (refer to the official KVS website https://kvsangathan.nic.in/) and the information given on this in the previous responses are incorrect.
There is a Departmental Exam for which only PRTs with five years of regular service are eligible to apply, so right off, as a primary teacher, you are not eligible since you have only three years of experience (you can check if there are any new rules currently, though). Also, those appearing for the Departmental exam must have a Bachelor's Degree in the subject/combination of subjects (refer to details below). While there is (a) age criteria of maximum 35 years and (b) the requirement of min. 50% marks in the Bachelor's Degree exam for direct recruits for KVS TGT posts, these are not applicable for promotees who already have five years of regular service.
Now let's say you do become eligible for a TGT post in a KVS after completing the five years' requirement, your options are to take up the subject that you did in all the three years of your B.A. Degree course. However, as per what you have listed, you have not taken up the required subject or the required subject combinations. You did not take up Hindi in the second year and English in the third year. You should have stuck to at least one of these languages for all the three years, which would have then made you eligible to get promotion as a language teacher in one of them. Details are taken from the official website -
1. TGT (English): English as a subject in all the three years.
2. TGT (Social Studies): Any two of the following - History, Geography, Economics and Political Science, of which one must be either History or Geography.
3. TGT (Maths): Bachelor Degree in Maths with any two of the following subjects - Physics, Chemistry, Electronics, Computer Science, Statistics.
4. TGT (Science): Botany, Zoology and Chemistry.
5. TGT (Hindi): Hindi as a subject in all the three years.
Similarly, there will be subject-combination requirements for other teaching vacancies. Same would be the case when seeking promotion from TGT to PGT - there will be a Departmental exam and subject-specific requirements of the academic degrees. Additional requirements are to be able to teach in both, English and Hindi medium and have computer knowledge.
Regarding doing your B.Ed., while having a B.Ed. is good for a teaching career, again you need to find out if the subject in which you do it would be beneficial based on your earlier academic Degrees, both B.A. and M.A. Let's say that you complete your B.Ed. in Hindi and then find that because you did not do that subject in one of the three years at the Graduate level, you are ineligible for a teaching post, then pursuing the B.Ed. course
would be a waste of time. You have done your D.El.Ed.
so find out if you can teach Hindi in higher classes up to Class 8 even without the B.Ed. and even if you have not taken it up in the second year of B.A.
Also, note that while a B.Ed. acquired through distance education may be acceptable in some Government schools provided it is from a recognized University, it may not be the case with all. In fact, I have been putting up teaching vacancies in the jobs' section
and have noted that in many private schools they are very emphatic that they will not recruit candidates with any qualification through distance education. You have even done your Graduation and Postgraduation through the distance learning mode, so not sure how this would be helpful in furthering your career as compared to doing a regular Degree course.
I would advise you to consult the School Principal where you are currently teaching and also PGT and TGT teachers of other schools who already have an established teaching career to get advice on your future career opportunities. I am not sure, but maybe the nearest KVS Regional Office which assists in the KVS admission process for students would also be able to guide in providing the information that you seek. Further, I also advise you to improve your English skills, both verbal and written, for better prospects in the teaching field.