ANH troopers, as you pointed out, can carry a DLT-19 and still remain canon. So yes, it's permissible.
As far as being approved w/o an E-11 and holster, that'll be up to your local GML. I'd still advise getting the E-11 and holster, which could be as inexpensive as a painted Hasbro with a second-hand holster.