if you transfer when the dirt is dry it should pop out the plug, set it gently into another pot preferably your final size and back fill gently with dirt, dont pack it or play with it, just pour dirt in it. Thoroughly water it till you see some run off and let it dry out at the bottom some before you do it again. I never seen plant shock from a transplant done this way. you are not root bound, you just cant keep up with the roots; demands in this little cup. A bigger cup will provide some buffer for you. I've seen trees grown in little cups, with roots trailing for feet out the bottom, as long as its watered and fed correctly it will thrive. strange though I never seen it get big around just tall.
yours look good !