It depends on how much you need the job right now and how fast you can find another.

But, if that commute is a dealbreaker for you I'd simply tell them that you had accepted the position because it's permanently remote and ask if it's possible to continue that agreement due to your commute distance and because that's why you were able to accept it in the first place. Just be sure to remind them it was proposed to you as permanently remote.

Then if they give pushback trying to say they can't honor it.

Say that you understand, but you will need to resign if that's the case. They may give in at that point, but maybe not.

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If you really can't afford to lose it, then I'd just start looking as you said, and try to bare it for a while.