Shadow Lair team build idea: Ice Queen
The team consists of 1 Tank and 3 DPS.
* The TANK:
Armor: Ice queen set with Pierce max UVs (alternatively mercurial set pierce max, because freeze isn't that bad in teamplay and MSI is very good)
Shield: Royal jelly/Aegis/Ironmight plate with a freeze high or max UV, otherwise Royal Jelly shield is probably best.
Trinket: Penta heart pendant
Sprite: Seraphynx, probably using valkyrian aura to get extra vortexes off and grab as many slimes as possible
Weapons: Electron Vortex or Obsidian Crusher (CTR High, lvl 10), Winter Grave (CTR High, lvl 10), perhaps Venom Veiler
* The DPS:
Armor: Black Kat set (Possibly trade 1 BK piece for a pierce max ice queen or skolver piece, using the Liquifier sprite perk to max out against slimes)
Shield: Royal Jelly shield
Trinket: Penta heart pendants
Sprite: Seraphynx is great to assist with heart attacks (your tank will need it), but maskeraith quills or drakon frenzied firestorm can be useful too
Weapons: Winter grave (CTR High, level 10), Venom veiler or vortex for additional support, further weapons are up to preference (e.g. Acheron/Grim for certain arenas, WRH for lumbers and Silkwings, Dark retribution). Damage bonus against slime UVs (high+) can let you drop a piece of BK in exchange for a piercing armor (like pierce max ice queen mail), combined with the Liquifier pet perk you could use a full defensive piercing set while still having max damage against slimes.
The idea is simple. The Tank will be in the fray placing vortex bombs to group up all the enemies. He can stay alive through his strong shield, strong piercing defense and poisoned enemies (DPS can place venom veiler while Tank is vortexing). Sera's aura can eat a stray spike from a polyp. After placing 1 or 2 vortexes, you should have a big group of enemies set up for the heart attack + winter grave combo. There are no particularly fast enemies, so there is nothing there to punish you for charging Winter's Grave.
Shield choice:
Not every attack has a freeze component in the mission. In fact giant lichen colonies are arguably the biggest threat and a shadow or elemental shield will not do well against them. The other big threat is polyps which do have a freeze component (but also piercing). Winter Grave is the best weapon to deal with polyps though, because you can stay at range and dodge their spikes fairly easily. Giant lichen colonies can close in with their dash and deal pure piercing with their spin attack (which also deals stun) and thorns attack; against this the best shield will be a piercing shield.
Winter Grave:
I'm glad you bring up Ash of Agni, it's a very nice way to thaw the ice cubes and weaken their attacks significantly. The problem is Winter Grave will overwrite AoA's minor fire. Iron Slug isn't a good alternative for Winter Grave though, because Shadow lair enemies have a lot of hp and defense and therefore the difference between normal and shadow damage is large. It'll probably take 1 fire tick for every magnus charge hit to make up for the difference between Iron slug and winter grave. So if a charge hits 4 times that is 4 ticks of AoA fire to break even. If 4 people are charging their magnus, it'll add up really quickly, but you can't make AoA fire tick any faster.
Another strong contender is Grim Repeater, but it just can't beat Winter Grave charges. If there were enemies that could disrupt Magnus-charge spamming, like greavers, then perhaps there would be some competition for WG. But with only slow jellies and lumbers to worry about, WG just beats everything. It is kind of sad for AoA that WG not only beats it but has anti-synergy with it in one of the only missions AoA is good.
your shield choice is wrong.
for defense against attacks with a status component, status resists are far more important than defense and a UV is nowhere close to giving as much as an innate resist does (which is hard to quantify, but definitely over 8.5 points, so at least double).
this means you should pick dread skelly (inferior to most others on this list but more likely to be already owned), heater (best choice in base gear), gorgomega (stun resist means more durable against silversaps), omegaward (if you have the choice, I would either pick gorgo or heater instead) or teddy (better than gorgo since it has no MSD, but is promo gear).
in terms of weapons, get one person to use AoA to thaw ice cubes and dont use winter grave to not refreeze, iron slug instead which also cuts through silkwings. also if you want poison, obsidian edge deals strong poison quite consistently.