That word 儘 is hard to grasp; I think it doesn’t have a direct equivalent.
But, yes, here (particularly with その まま as you found out), it is about the idea of a continuous unchanged thing.
I think the idea is that,
そのまま死なせてやる
decomposition: that まま / to.die+causative+do.for.him
=> make him be able to die without changes
=> let him die peacefully
(For what it’s worth, Google translate gives “Let me die as it is”)
And if we look at the translation made in the wasabi site (I put the link here), they translated the whole sentence (どうしてそのまま死なせてやらなかった) as:
“Why didn’t you just let him die?"
そのまま rendered by “just”.