If the tank rotates his mitigations instead of using all of them at once before the damage hits, then he's probably doing fine. Ideally, he would not be using invuln first since it becomes wasted with Holy's usage as stunning an enemy is the same as not taking damage. Even then though, you shouldn't be going out of mana unless you haven't used Lucid Dreaming at all.
Your healing rotation plan - oGCD usage first (using Asylum, Tetragrammaton, Benediction, Assize first before going to use regen/lilies and then Cure II since the other skills don't cost MP). If a WHM holds off from using their oGCDs first until an emergency happens, they're basically doing what a bad tank would be doing: holding off from using mitigations until their HP is almost gone, and then panic spamming their mitigations to survive without taking into account of future damage.
You should be able to get at least 1 Holy off if the tank uses mitigation even in a big pull before the tank runs out of HP (because if the tank can't survive before the first holy cast, it means he died under 3 seconds and that means something else is the issue here instead of just too much adds since tanks aren't that squishy).
Between the First Holy stun and your next Holy (4 seconds stun), There is time to apply a regen/lily + weave in 1-2 oGCD abilities. This is because holy's stun durations do not overlap. Holy finishes casting in 2.5 seconds, but the actual stun component and damage hits in 3 seconds. That means the first stun doesn't go away within your next GCD since that's in 2.5 seconds. The second and third stun could be chained together though.
If you feel the tank is losing too much HP in the first GCD, you can choose to apply regen/lily -> Swiftcast + Holy -> oGCD healing (Tetragrammaton/Asylum/Benediction).
If you can't get a single Holy off, then either the tank didn't mitigate or the adds didn't get interrupted before then because it means the tank had to die within 3 seconds from 100% HP, which makes them as squishy as a DPS. If that was the case, it's not your fault.