You can tank hami in melee. Everyone on Everlasting who tanks hami, including myself, does so in melee, and it causes no problems. It makes it easier to do without outside buffs as well because you can use more self heal powers, such as DNA Siphon.
While the league is clearing mitos, no one should be close enough to you to be hit by the aoe. If necessary, you can pull away while the league fights hami, but in my experience, this is not necessary.
But yes, the best builds will have: 1. high max hp, 2. high regen, 3. reliable self heals.
Regen brute, bio tanker, dark tanker, and probably willpower tanker are all good picks. Make sure to get all the accolades that buff your max hp, vigor core alpha for better regen/healing, and melee hybrid for even more regen (once again, improved by being in melee).