Is your mage using remote servant to add swarmpets on egg ? if not there is your problem.
Have remote servant loaded, stand outside web, target egg, cast and place the green circle for remote servant outside the room
Servant spawns and runs in attacking egg
Are you buffing your pets with gear ? e.g giving them at least legs with ferocity on it (I use dimensional warrior greaves/conflagrant warrior legs)
You have to give pets weapons, even shm/ench pets
I am thinking of all three being the same class and using Alliance.
Alliance is a spell and needs to land, if that was possible your wiz/mage would destroy those eggs in 2 casts.
Doubt your alliance idea would work.
Another trick is mage (bolt spells) can hit spider and eggs through the floor if it is one of the top floors, maybe wizard (beams?) too, never tried with a wiz.
So maybe leave mage in one of the lower rooms
That said I do think you have enough pet support
I get the achievement with
sk/shm/bst pets but then I do have a bard that adds procs/dps for pets.
You have enchanter/mage/shm, should theoretically be enough.
If at the egg a DPS can use all it's abilities to fight it, it should be sufficient to destroy it in time.
Only works in the northwest lower room, if melee dps attacks eggs in other rooms they get ported out.
If you are really set on making another character, make either a mage/ranger
Mage can add pets, ranger can hit 3 of the room eggs from outside the web.