ざるを得ない：cannot help (doing); have no choice but to; am compelled to; am obliged to; it is incumbent upon (me) to

Is another proper translation そう考えない訳にはいけない？ (Checking my understanding from this sentence: 私はこの女性労働者達の健康について心配しないわけにはいかない。)

Also, is this a common Japanese way of expressing this meaning? My understanding is that ざる is an archaic construction.

This altered sentence sounds unnatural for two reasons. You need to stick to the original Japanese sentence.

First, わけにはいけない is grammatically broken. It should be わけにはいかない.

Second, as I explained on the SD on わけにはいかない, わけにはいかない has the connotation of “obligation” or “commitment”. In other words, if the speaker didn’t think this way, the speaker might be punished or penalized when you say そう考えない訳にはいかない.

However, “I have no choice but to think this way” in English doesn’t have such an obligation or commitment. The nuance of そう考えざるを得ない is closer to “it is inevitable (or reasonable/natural) to reach the conclusion of this interpretation”.

～ざるを得ない is a set phrase. It is still widely used in modern Japanese although ざる (negation) alone is a little archaic.