Idea: Beast Master Tank

[Sylvain]

Weapon: 2h Hammer (though dual wielded hammers would also fit)

I would see dual 1h axes but that's just me

Shared pain - When master/pet takes damage equal to 10% of max hp, damage is increased by 5% for 10 seconds, stacking 5 times and a maximum of 60 seconds, at 5 stacks gcd is reduced by 15%

Well I see a core issue with this, you need to take a set amount of damage meaning that once you're tanky enough, you can potentially loose the 25% damage done. On top of that it would be the only tank that do more damage by tank instead of off tanking.
Basically, you get diminushing return for being what you're suppose to be, a tank.

Lick wounds - when master/pet deals damage equal to 10% of max hp, begin to regenerate 1% of max hp for 10 seconds stacking 5 times for a maximum of 60 seconds, at 5 stacks, heal for 20% of criticial strike damage.

Same issue than the above, if you don't get dmg you don't get 25% more damage so you deal less damage and proc Lick wounds slower.
Altough I suppose 10% of your max life shouldn't be any hard to deal unless they drastically change the hp/dmg ratio. Atm it would take 5 gcd to stack.

Main Mechanics 2: Challenge(master) / Warcry (Pet) - stances 30 second cooldown
snip

Well first i'm happy to see you went beyond the whole "my pet tank the whole time". Considering how pet are unresponsive in this game this would be extremely unpractical

The concept of making the boss go back n forth between you and pet is interesting.

There's however a glaring issue with this. First, unless you specifically make that any debuff on the BSM is also applied to the pet and viceverca, BSM would be able to solo tank most boss requiring regular tankswap. I'm sure I don't need to explain why this is bad in term of design in an mmo.

second, 2 tanks mean HoTs and shield are twice as effective. I'm sure this could be easily exploited. The fact that you can swap your hp bar at will every 30sec is imo way too strong. Again you would barely need any heal (i mean, even less than tank currently do)

Finally, being able to sacrifice your pet on select tankburster without loosing the player is obviously extremely strong and would be exploited. Unless the penalty is as bad as death, you can bet you would see a lot of pet sacrifice strategy in order to do a boss with only one tank.

The master/pet would have 70% of the total health of the other tanks, because they would be expected to swap back and forth for the full hp, and there would be some penalty in place for the master for letting the pet die.

Finally, I may be wrong but reading this makes me feel you havn't done anything beyond dungeon and regular raid.
Unless your tank would have a crap load of strong CD to survive tank burster as well as a regular tank would with the same strong CD, having 30% less max hp, with the same kind of CD as currently available, would result in very hard to survive tank burster in savage. Just byako which isn't even a strong boss can easily burst throuh 60-70% of a tank hp under rampart on his tank burster.

fun concept to flesh out

Actually it would, but I don't think making a pet actively tanking (beside special cd where the pet takes the next hit for instance) would work in FF14 due to how raid are designed.
However, the way you putted it sounds more like what a tank would be if heal was scarce and hard to get.
But it isn't. Beside tank bursters, it wouldn't be remotely hard to keep you or the pet alive so your swap aggro thing would be bordeline useless (the same way all 3 tank always tank in dps stance).
Unless you're about to die (or the pet), you would never need to swap aggro and that seems to go against the core fantasy behind your job.

It was a fun read.