Totally, 100% soloable.
I've got all the pets from part 1 except 2 from BWL solo.
I've got one of the pets from Part 2 solo.
They do seem to have nerfed the drop rate slightly, as I whizzed through part 1 and have had 2 dry weeks on part 2 (except the Big Bad Wolf).
You are better off doing them solo, more chances for a drop!
EDIT: I also did them over a time frame. Got most first week but it took me 3 kills to get the Twin Emp one, and 4 to do Viscidus on the DK (during which time I never got a duplicate from the prophet
). I also only recently learned to solo BWL and got one pet the first week, that was a week before 5.3.