Tried hidden menu?
when you can you should try this - it might work for you:
In dialer enter:
That should give you a secret menu. Select Phone information, go to (almost) buttom and set preferred network.
If no menu comes on you phone try download "Secret Codes" app from Play. There are more codes that may be relevant for you model.
Why: Phone may be capable, but GUI does not get the setting through to you
How I found out:
I had no initial problems with LTE on stock. Flashing 4.4.2 roms ****ed that up a bit. Option was there but not steady and other problems.
I am now on MIUI which is JB (but fast/stable/funny) and that rom misses the option for LTE as you discribe. First time on ROM it connected fine, but at a time I was in an area with no coverage, and by mistake I changed the option to 3G from settings.....
I used the secret code, and connection is stable. Only had issue once, but now know to change