tbh i feel like it's actually the opposite - or in other word i agree with
@Mike852605, if we're going by examples by kpop artists. take Shinhwa for example, the members left their agency and have to fought some lengthy legal battle to keep their name and to name their own agency "Shinhwa", or the more recent T-ara. all the members left the agency and the agency patented the name so they wouldn't be able to use the name T-ara anymore and the members are now persecuting this matter so they would be able to use the T-ara name for their future activities. same case goes with b2st which have to change their name to Highlight.
If Keiko's the one who leave and decided to follow Kajiura, it would make sense for her to apply for Kalafina's name, seeing the name was made by Kajiura so that the remaining members/the agency couldn't use the name Kalafina anymore for their future activities. the members who're staying with the company could just left the name/patent matter to their company and wouldn't have to apply for the patent themselves.
if the company intends to keep using the name Kalafina they should be the one who apply for the patent/right to use the name seeing as they're the one who will be managing the group. applying for the right/patent to use a brand/name could mean 2 entirely different things, to be able to keep the right for using the name and to prevent other people to use the name.
tldr: afaik in most cases the ones who're leaving are the ones who apply for the right/patent for the name of the group, before the company applied for the right/patent of the name.