One option is that they simply blocked the OP from working, and without it Lanfear and Moiraine would both have been utterly helpless.

Remember, that the finns had in their possession a ter'angreal that had the ability to completely nullify weaves of OP. They gave it to Mat. We also know from BS that they have a very large collection of other ter'angreal, most likely thanks to their dealings with the ancient AS.

I mean for all we know, they may have hundreds of versions of amulets that do exactly the same as Mat's foxhead does. Or they could have a copy of the Far Madding Guardian ter'angreal that they can activate in emergencies to completely disable all OP channeling in the area.

Remember, these guys KNOW the Aes Sedai and their powers. They made bargains and negotiated with them. The Aes Sedai likely made the doorways into their realm in the first place. It is not all that unbelievable that the finns would have anticipated the possibility of a channeler (Especially a shadow affiliated one) openly attacking them, and prepared for the possibility. Heck, the ancient AS may have even been happy to help them with that, to prevent the shadow from forcing the finns to aid the shadow against the light.

And at the end of the day, even if we ignore ALL of that, Lanfear was trapped in their realm. Sooner or later, she would have grown tired from channeling or had to sleep. And then the Finns would have had her. But I think it more likely that the finns simply used somekind of ter'angreal to disable her abilities very quickly, since I figure otherwise the first thing she would have done would have been to burn Moiraine into Ash.

Finally, we don't know if Lanfear got any wishes, or if she was protected by the treaties. It has been mentioned that questions that touch upon the shadow can be dangerous, and potentially allow the finns to attack you, even if you came through the doorways. The original treaties were also negotiated by the Aes Sedai who opposed the shadow. It is not beyond doubt that the treaty has a loophole in it that allows the finns to do whatever they want to anyone who is a darkfriend. Moiraine however, would have gotten her wishes since she is NOT associated with the shadow and as such bound by the treaty.