Protestants argue that the bolded part means divorce is moral in cases of adultery, and I'm not really sure how to refute them. What exactly does this passage mean - what's the exception here actually about?
Close but no cigar.
It doesn't mean you can divorce and remarry. It does mean that if your spouse commits adultery, you can refuse your spouse the marital debt for the rest of your marriage (until the death of one of the spouses), and have a separation of bed and board ("live in separate houses") in the meantime. And yes, nowadays that would require filing for a civil divorce.
But be careful! Once you know your spouse carnally after the adultery was discovered, that's considered taking them back or forgiving them, and you can't change your mind later. Unless they commit another act of adultery -- then you get to decide again.
But for Catholics, there is never question of remarriage, or getting another chance at it. The intimacy of marriage was not meant to be shared with 2-3 (much less 50+) people over your lifetime. God designed marriage and the marital union, and He knows all of its effects on individuals as well as society as a whole. It is fitting that God made some rules to govern its use.
When a woman (or man) catches their spouse cheating, there is a choice: reconcile/forgive them, or not. One does not have to. God, being the Author of human nature, and the Church, being a good, understanding mother, knows how difficult/impossible it can be once trust has been broken in this way. So
God doesn't make it impossible for us.
On the other hand, you don't get to divorce, ever. So while you can separate from your spouse (even if you suspect they will leave the Church, get remarried, etc. -- that's on them) you have to plan to live out the remainder of your life celibate (single) afterwards, unless you luck out and your cheating spouse gets hit by a bus!
Be careful and don't rush into marriage. I know some have argued that technically there are no guarantees, and that is true. However, in MOST cases, those who married poorly can only blame themselves. They could have avoided their fate if they had been more objective, had their eyes open, listened to (had more respect for?) their parents, and so on. Do your best, do your due diligence, and leave the rest in God's hands.