The sentence appears to have been cut off at the end.
You need to add a verb at the end, like so:
"오리지널 멤버가 아니기 때문에 2기 꼬리표까진 어쩔 수 없겠지만 그렇다고 다른 이름은 생각조차 해본 적이 없다"며 "베이비복스의 전통성을 제대로 이어 나가겠다"고 (말)했다. (Also, 전통성 is probably a typo of 정통성, although 전통성 can also work as second choice).
=> Saying "she can't do anything about the 'second generation' label as she's not an original member, but even then she has never once thought of adopting another name", she vows to "continue the identity of Baby V.O.X as best she can".
* 어쩔 수(가) 없다 = 어찌 할 수 없다 = 어떻게 할 수 없다 = can't do anything about it. Can't help but accept it as is.
* 그렇다고 = 그렇다고 해서 = just because of that (it doesn't mean ...). (what was just mentioned) doesn't mean that ...
* 생각조차 해본 적이 없다 = have never even thought of doing it (not to speak of actually trying it - an emphatic denial).