I say the absolute most important thing is that You and your family are happy with your school. Regardless of which Org, if any, they belong to is irrelevant at this point. If you are just beginning and are happy with everything that is the #1 goal.
Belts recognition and transfer recognized by ORGs is usually only relevant at the Black belt level. Most color belts are beginners and have not been "molded" yet so if you do move, most schools wont have an issue with which color belt you are unless it is a higher one near or at Black.
The type/style of Forms you do will be the factor of belt transferring too. I wouldn't get too wrapped up in the transferring of schools and belts unless you plan on leaving. but even then, it will depend on the school you are going to and what they are doing.
which forms do you do?
I still say the
MOST important thing is you and your family are happy training at your school.
As for the Ji Do Kwon federation, I would not be the subject matter expert on that. My generation, does not care about the original Kwons and preserving that heritage. As far as I am concerned, the 5 Kwons unified for taekwondo and formed Kukkiwon when I was Born and before I started training. So as far as I am concerned, Its ancient history
But seriously, there are some who still preserve the Kwons. Punnui could give more answers on this area. He would be better suited to answer your Kwon Specific questions.