For Mummy I have the BB3, the three Sekhmet BB's and then Scarlet Ladies as her only special move. Only one because if she's got more, the chances of using them too soon after one another (i.e. before the Heavy Regen from the first one runs out, thus taking off a second 20% of her health, but this time without giving her the 10 second heavy regen to make up for it) are really big when controlled by AI, which is exactly the "problem" you describe with the Eliza on Peacock master.
I'd been thinking whether she should have Scarlet Ladies or Chaos Banish, but I felt that it was most important that she gets back her lost health (and more, due to SL giving another 12% health with regular regen on top of the needed 20% from the Heavy regen) as quickly as possible. With this it only takes 7 seconds for her to get back the 20% instead of needing the full 10 seconds.
The three Sekhmets for the Blessing they give, as well as being I think the core of Eliza's moveset on offense. The BB3 for obvious reasons in this situation.
I have a lot of special cooldown so in reality she can reuse SL after 7 seconds, but AI's usually don't use the same move that quickly right after another + the chance of the situation being so that exactly in those last 3 seconds neither fighter is in a combo, especially so shortly after her getting 50% blockbuster meter, is VERY slim. And I don't think many players will risk letting her get another 50% BB-meter and therefore her BB3, just to have her take off 20% of her health (12% of which WILL be regenerated anyways due to her getting the regular regen regardless of timing).
And on top of all that is that all the Sekhmets + a decent amount of meter gain makes the possibility of her killing herself with her sa even less of a problem, because she'll most likely have at least one stack of Blessing anyways when it happens, letting her at least stall JUST a bit more time, but maybe doing more if there's 3+ stacks of it.
(Yeah I wish we could fight our own nodes as well. But that might also maybe put TOO much power in the hands of the rift builder, and take away the last bit of guessing/personal flavor, reducing the whole process to what is essentially a sort of trial-and-error math problem. That would really take the fun out of it tbh)