Chulalongkorn Go in Seacon is not an integral part of Chula Language Institute. The link appears to be tenuous.
To my knowledge, no farang teachers from Chula teach at this place.
I heard a rumour that one, or maybe two, of the Thai teachers in Chula have a private interest in this place as private individuals. I am unsure whether this is true. Even if it were, then that is their business. I have heard that the place is well-run and that the salaries are good.
Some may find it contentious that the place uses the name, and suggests an association. I am unsure what the law is re. copyright of names etc. Perhaps anyone can open a Mahidol Bookshop or Thammasat Coffee lounge.
If you applied for weekend work, you would not be working for the university. Chula lecturers need to have (at least) a master's and considerable teaching experience in addition to CELTA or DELTA. The selection procedure is quite rigorous, as you would expect, and includes a three-hour theoretical paper as well as a teaching demo with a full class, and panel interview. Certificates, transcripts and references are always checked, and you will not be allowed to work on a tourist visa or any other dodgy basis. If you fit the bill, which you may well do... you may wish to apply for a university post, but the two places are quite different. If you became a Chula instructor, you would work a normal week, but there could be the possibility of some form of weekend work from time to time. I suspect also that the client groups (students) are dissimilar, with more younger kids at Seacon.
Good luck.