Hi tjs...and welcome,
I know that Elvis did own a number of dogs throughout his life and career, but I'm not very good at naming the breed of them.
In "Jailhouse Rock" there are a couple of basset or bloodhounds (not too sure, but same breed of dog that Elvis sang to on the 'Steve Allen Show') but I don't know if they belonged to Elvis or not.
If you've seen his 1965 film "Paradise, Hawaiian Style" and the obvious scene that springs to mind is in the helicopter with a number of dogs, I do believe that either all of them or at least 2 of them were his pets.
And, of course, who could forget the 1968 movie "Live A Little, Love A Little" which co-starred his Great Dane, 'Brutus' (known as 'Albert' in the movie)...I do believe that 'Brutus' had a partner and her name, I think, was 'Snoopy' (?) and she was known as Lisa-Marie's pet.
And, finally, as you mentioned there was poor 'Getlo'.