Did you try equiping the exact same type of faction armour?
As when it got stuck for me the one time I tried every type but only the specific faction one worked ( in my case NCR. )
If you had another factions gear on previously before the white gloves try it with that instead.
Tried it with every kind of faction armor I can remember wearing -- no luck. Also tried clearing my 360's cache and re-downloading the patch -- nada.
It definitely seems to tied to the White Glove clothes specifically; every time I put them on, a faction disappears from my list. One by one, any faction that has faction clothing goes. Put it on; bye, bye Caeser's Legion; take it off, put it on again, bye bye White Glove Society. I can only imagine that the NCR, Powder Gangers and Khans all went down the same way, one by one, when I was wearing the White Glove Attire before I ever even noticed the problem. But that was hours and hours ago and any saves from before that time are long gone.
*Sigh*
At least I'll be able to build my character slightly better this time around. T_T
Thanks for the help, though!