I want to double down on that point about shaman pet. They just aren't bringing enough to the table to begin with to be amplified enough to be worth it (in my opinion). Every time you or your pet dies, you lose everything that was on it in an instant, and shaman pet just doesn't provide enough to justify that amount of work, to me. That said, they are much beefier than they were in years past, so I wouldn't say there's no situation in which it would be worth it. There isn't a situation where I would find it worth it that I can think of, however, because I'm never in a situation where the shaman pet taking some hits would be a saving grace and the little extra dps just isn't a make or break for me. I've got plenty of dps.
I personally wouldn't do it to the SK pet either for the same reason, but arander has deemed that worth it, and that's a point worth noting. In their case, they have found it to be worth it just for the sake of not needing to bother with redoing the pet. At the end of the day, if you find it worth it to you personally then that is justification in itself.
So, if I have a BST pet and I’m looking to max dps, what should I do?
I'm sure the reason arander is using conflagrant specifically is because its ease of attaining and it's relatively good stats. - If you truly want to maximize dps and ease of replacing the piece on your pet isn't a factor, you want whatever the best attuneable piece of gear you can find is. ToV T1 non-visibles are attuneable and quite a bit better than conflagrant, but you need to actually be able to get the items. That said, the heroic stats aren't significantly better, so you shouldn't expect the pet to become significantly better at dps. It will gain quite a bit in the way of tanking, though, from all the extra AC/pures.
Conflagrant is player made, so arander would have no issues replacing any piece so long as they have plenty of TS materials. The fact that heroic stats have gone through the roof because of augs not the stats on the items themselves means conflagrant is a great happy medium between accessible and powerful. That said, if you can get ToV non-visible T1 pieces for dirt cheap in the bazaar on your server, those might be worthwhile to snatch up for your pet (prices will vary since it's good non-prestige gear). Or run the ToV missions you can burn through in no time to get the chest loots as often as possible.
Purely for dps, heroic dexterity and heroic strength are the stats you should see the most benefit from, so just find whichever pieces favor those.