Kadaj regarded Ignis. Phoenix was probably a good fit for him. If nothing else, Ignis seemed to be taking the seriousness of Summoning to heart, even asking for Tayuya's advice. Tayuya's would probably be more useful to him; she knew the Fayths and their locations better than he did.

"So when do you want to go?" he asked. Grimmjow and Yylfordt would probably jump at it, and they hadn't taken Apache yet, so she might enjoy that.

Not the fire cavern so much. Kadaj would need his own protections to get through there.