Well, there's really no simple answer to this question, but that also implies that anything will work if you set it up right! How that for not being helpful at all!
You're right in my wheelhouse here as I'm a big fan of the stoner & desert rock stuff. Hard not to be...Sheavy is from here! In fact their new guitarist is a pal of mine.
The simple answer is it doesn't really matter as the boys use such a wide variety of gear. I've got Kyuss clips with Josh playing through everything from what appears to be JTM45 half-stack to a Plexi Bass 100 through what appears to be a 8x10. He plays Ampegs these days (I think). Some of the dudes dig old Sunn bass heads. A lot of them generally play through the rhythm pickup to make things even thicker & sludgier. Some of them use natural overdrive (with the amps on cremate) while others rely on various fuzz & OD boxes, or a combination of the two. I don't know if Mike Amott (Spiritual Beggars) plays through a Dual Rec, but it sure sounds like he does on some of the live recordings.
If I were going to choose a Trinity as a baseline, I personally would go with one of the 18w variations along with a boost and/or fuzz depending on what type of tone you wanted. The Triwatt is a little too "hi-fi" in general, but I did have it copping a pretty cool Fu Manchu fuzz thing with my SG straight in with the channel vols & OD way up.
BTW, I suspect the guitar in the Sleep vid is running through the Laney AOR. That tone sounds very Laney to my ears. I know they boys were Sabbath junkies also, so I wouldn't be surprised if there a Dallas Rangemaster Treble Booster pushing it.