One of the first things I thought of here, which may not be in the spirit of the question, is how many albums? I think variety would be key to not going absolutely batty.
Turns out W.A.S.P. and the Barenaked Ladies have about the same amount of studio albums, 14 or 15, so that's not a big factor. What comes out of this is, holy shit, W.A.S.P. made that many albums?
The answer is Barenaked Ladies. The first song that came to mind when I saw this question was Call and Answer, one of my favorite singles of the past couple of decades. That alone tips the scales. Also, as body of work goes, the Barenaked Ladies have the next eight or ten best songs.
It's pretty much a landslide victory except for the fact that W.A.S.P.'s singer, Blackie Lawless, has the coolest rock 'n' roll voice ever. That gives me pause and forces consideration where it's hardly necessary. Problem is, I think listening to W.A.S.P. for too long would just make me want to listen to Motley Crue, and on this island scenario, that would be infuriating. The Barenaked Ladies don't really sound like anyone else, and that would completely eliminate the problem.
In case I need to make more of an argument, the Barenaked Ladies are by far the more decorated band, and have a huge lead in sales, awards and Billboard success. The quality and variety of their catalogue wins out.
To be more accurate in terms of critique, I don't think W.A.S.P. has the riffing chops to be up there with other great hair metal acts. They have a comparable sound to better bands, and that just won't do. Better lead guitar and more dynamic drumming choices would serve them better. Even on their big hits you can here this complacency in their song-writing, a sickness in the Los Angeles rock scene in the eighties.
In fairness, W.A.S.P. has made a lot of music over the years, and they've had some pretty talented members. And they wrote the songs Blind in Texas and L.O.V.E Machine which are pretty badass. I think like a lot of bands from that scene, they would've been better served in another place and another decade.
Short Answer: Barenaked Ladies. 'Cause some days you don't want to be yelled at about pussy and booze.
Note: I didn't choose W.A.S.P. because I was angry at having to type W.A.S.P. over and over again. It's a real pain in the ass. I assume you did this on purpose to fuck with me. Way to skew your own experiment, obvious member of the band W.A.S.P.!