I'm no Warrior expert, though judging from some of the posts above your Warr clearly has a few things which could be improved.
However, as a Holy Pala I find it hard to understand how during a few of the deaths your Warr goes 4-6s without receiving either a direct heal or something through beacon from your Holy Pala. During cat phase there is nothing to heal beside the tank and whoever gets leaping flames (unless the orbs are up), so at worst there should be something going through beacon from the lf target, if not direct heals. If you are moving in at those moments a hpala has 2 sprints and there should be no way in any case that it should be taking 6s to get into position, on one your tank receives 5 ticks of Holy Radiance without receiving a single other heal from the hpala, what is he doing during that time (around 4.2s) if not casting any heal either directly on the tank or that transfers through beacon?
Not to just pick on the hpala, but being one myself it's easier for me to do so, shortly before most of the deaths he only receives a heal from the druid most of the time. Certainly the tank is at some fault judging by other comments, but at times it seems like he's not getting as much help as he could be from the healers either.
As to why the healing is suddenly fine when you switch tanks idk, maybe the healers struggle with the more steady incoming damage on the warrior, rather than the spiky damage on a bear, but "he takes more damage" is usually the first excuse I go to when I was casting a holy light thinking of my mana and the tank dies because it should've been a divine light And with that said perhaps a 2nd trinket with a spirit proc/on use would be a better choice than the Alysrazor one if he is having mana problems, likewise with Heartsong v Power Torrent, I had no mana issues on this fight (we had a RSham, but no hymns or innverates), with plea during transitions probably 3 times per try it really shouldn't be a problem.