I think this is a variation of ておく - to do something in advance; or to leave something in a certain state.
You can use causative verb + ておく to mean: When you leave someone do something they like/ let someone do something for a while intentionally. / leave some state the way it is intentionally.
When the verb already has a meaning of “to leave” or “to let someone do ~” or a causative meaning, you don’t have to use the causative form. Just use verb te-form + ておく ( = teoku).
In our sentence we have:
生かす - to let live
おけない - negative potential form of おく contracted to おけん
So I read it as - “we can’t let him live”
I’m not sure if the は is optional in this sentence and added to emphasise that this is a negative sentence. There is an example sentence in that Maggie Sensei article with that same てはおけない phrasing.