TJs naturally have a little play in the steering.
Your tie rod ends will rotate forward and backward, but they should not have okay side to side.
Your tires are very likely the problem. 45-50mph is very common for there to be a shimmy. Out of balance tires or bad tires is almost always the issue. I know you had them balanced recently, but did you get a perfect balance, especially when running bigger tires? Did you talk to the tire tech and make sure he understood that our TJs are very sensitive and must be balanced not good, but perfect? Most don’t know and won’t take the time to get them perfect. What kind of balance machine did they use? Static or dynamic road force? How old are the tires?
I’ve had this problem and had multiple balances at multiple shops only to have it solved by getting new tires.