Bring in some lycanthrope hunters
You've noted that lycanthropes are not common on the surface. Because of that, I suspect that most potential enemies (both good or evil) will not be prepared to face a lycanthrope. That makes sense, and letting them just have this for a little while will make them feel like that award was substantial (it is!).
But... even if lycanthropes are uncommon, you have not said they are non-existent above the surface. And like anything that is different, scary, and not understood, there will be someone who wants it gone. It will be a known fact (or at the very least a half-believed rumor) that such creatures exist, and there will inevitably be some people who accept the "responsibility" of ridding the world of this perceived evil. It's not that much of a stretch to think that some of these people would have found each other, shared their knowledge, and become very good at removing (murdering) lycanthropes.
So, as the player characters continue through your adventure, eventually someone will notice how they seem to just shrug off deadly blows, or - gods forbid! - witness a transformation! And if (when) this happens, it's only a matter of time before word gets back to these monster hunters!
At this point, you now have the chance to introduce a deadly and persistent enemy who will hunt them for as long as they remain above ground! Your party might prove themselves equal to the task, or they might find this threat overwhelming. If the latter is true, they might even decide to take sanctuary in the lycanthrope city in the Underdark! I'm sure they'd be accustomed to receiving such refugees already, if such a deadly group exists.
This enemy would likely be persistent, refusing to simply let the party be. They would pursue your party to the ends of the earth, hunting them down like the animals these hunters believe they are. Your game would take on an entirely new element if you go with this, as your party now always has to worry about being tracked! It's also not unreasonable that if these hunters coordinate, that they would set up ambushes, or try to target the party while they're already busy dealing with something else entirely!
Let your Underdark enemies actually be prepared for this
If your party does choose sanctuary in the Underdark, they get a base of operations where they are actually safe from those who would hunt them. However, now they're in a place where lycanthropes are not just normal, but have an entire city. Any nearby enemies, be it mercenaries, bandits, or rival city-states, would be equipped and trained to deal with lycanthropes.
These wouldn't just have hodge-podge knowledge and whatever gear they can get their hands on to fight. They'd have years, perhaps decades, of experience dealing with and fighting an organized group of lycanthropes. They'd know better than anyone, and be better equipped than anyone, to kill a group of lycanthropes of any size! Your party may be safe within the walls (or borders) of the lycanthrope city, but the enemies outside of those walls are a greater threat to them than anyone above-ground would be.
This will probably change the tone of your game if you do this
Though I think this is a perfectly reasonable way that the game-world would react, choosing this option would almost certainly permanently alter the tone of the game. The party could never stay anywhere for long unless they just decide to lie low and not attract any attention - and even then they might be found out. They'd gain a lot of incentive to remain secretive, yet to somehow at the same time gain the locals' trust in case they are found out.
It would become difficult for them to stay anywhere for long once their nature had been discovered, as word would almost certainly get out and draw in hunters - especially as the party's reputation grows through the quests they complete. They'd have to spend much of their time looking over their shoulders, as the characters are literally being hunted by people who specialize in killing them.
On the other hand, this could lead to some very touching moments as a village or township stands together to protect these heroes who saved their homes and earned their trust even in spite of their unusual nature! Then you have to decide... just how far are these Hunters willing to go to catch their prey?
Personal experience with this kind of approach
I haven't used this approach before, but a similar approach has been used on me, in which our party was hunted by a shadowy group of assassins. The GM thought we hated it, but for several of us it was our favorite part of that campaign!
There's a certain thrill when any otherwise mundane encounter could contain a deadly threat, and we often found ourselves taking greater risks than we normally would if there was a chance we'd kill one of our pursuers. Like... lingering in a building as it burns around you to make sure that guy is dead! (I might or might not have been the guy lingering...)
I honestly don't remember a whole lot about that campaign itself, but I vividly remember how abruptly the tension would shift as soon as the knife flew out of the darkness, or one of these guys slipped out of the shadows! In every encounter with them, our priorities would be overturned the moment we knew they were there! Whatever else we were doing, it was at best equal in priority to eliminating one of these guys! Any scrap of information we could find on them was worth more than mountains of gold, because they were so dangerous to us!
Amazingly, we actually didn't lose anyone to these guys, but half the time it was a fluke more than anything else! Every fight with them felt like a fight for our characters' lives! The game was (at least for me) far more fun once we were actively being hunted by them!
I will add that the pacing of those encounters added to it a lot. Once any of these guys showed up, they would attack us relentlessly as if every blow was intended to kill. If they found themselves at a disadvantage, they would try to break away and escape so they could strike at us later. We occasionally found ourselves having to quickly decide whether they were actually on the run and should be pursued, or if they were trying to lead us into an ambush (both did happen btw). Any fight with these guys would begin with almost no fanfare and would often end just as abruptly.