Hana burns smaller ghosts simply because she's really powerful but bigger ones can consume her power. If we compare her to a fire, then the ghosts are "cold" and can take in her "heat". It's like how a fireplace in a house burns more slowly than a similar amount of wood in open mountains. The ghosts create a "cold" environment around Hana and suck in her heat, causing her to burn out faster.
I wonder if Julia already realized that Zen is bad news as well. She may be unable to see the "don't look" ghost but the smaller kitties should be small enough that she'd be able to detect them at least. She may finally help out with something because she at least has the careless chuuni attitude born of ignorance, so she may end up being the one to break Miko's pacifism and explain that she needs to act to protect Hana.
I don't think Miko would call in Inari's protection so readily. We see Inari as a summon scroll with three uses, however Miko seems to think Inari's a life bar with three lives. She may be right.