Well, as I haven't played much LCS yet (But am trying the change that right now), my suggestions can't be very technical, and might look out of place (such as already implemented, or just not consistent with the game).
So, the things I can put together regarding your questions about the mole :
1: How would we get one ?
Hmm. How would they infest such an organisation in real life ?
They could be fake liberal right from the beginning. Once things start to heat a bit too much for them they would go to the police.
Or once they are caught by the police, the police prefers to let them go in exchange of some info. They could have ways to put pressure on them. (We know enough about you to send you to jail you know. But we might make an arrangement...)
The moles would have relations out of the LCS (met before or after you recruited them). These relations could be used to put pressure on them. Or simply by being confident, they could give to their relatives too many informations.
So there would be many different ways. In terms of gameplay, do we really need to model that ? I don't think so. There could just be a random check on some of you character agains juice, heart, wisdom or intelligence (I don't know which characteristic would be best) to see if he becomes a mole.
Eventually the result of that check coul be stored, a given somewhere when we discover who is that mole (we would then know if he was caught by police, if he betrayed you unpurposelly by telling too many things to relatives...)
2: What kind of info would a mole leak ?
Well, you described a new police and crime rating system. Maybe by sending info to the police they could add a modifyer to someone which would increase its rating. Or worse, simply give info on the safehouse, which would reduce its 75% "heat absorption" to 0. (He simply tells that there is an hideout somewhere in that building).
How do we decide which info he gives : It could maybe be any info on any of its direct contact (higher or lower) (so as you told, that would be close the guyes caught by the police who speaks ?). But instead of losing all contacts at a time, they might be caught one after the other in their every day activity.
They could maybe give info on all the people they seen performing a criminal act (thus increasing the heating he would bet from that specific act).
They might steal things from your stock, or money. (you'd then be warned by a message.)
Eventually, they could just disappear one day.
They could give info of one of the squads moves (the one it is part of, or the one under its level...). When the squad arrives at the place, the police would be waiting for them.
Oh, a last thing, maybe funny : they could submit wrong orders to all their lower contact. Instead of doing what you told them, they might do something really foolish.
3: How do we know that we have got a mole ?
Hmm. Good question.
If the police is already on site when you arrive, that's easy, you know that you have been sold.
If the guy disappears or give wrong orders, the result is quite obvious too.
(In these both cases, in terms of gameplay a message can be printed "Rats ! we've been sold. There is a mole !")
For the guy who gives some info, it is really less obvious for the player. that maybe where police sleepers would usefull ("well that man has been arrested. We got some info to lead us"). Or in front of the justice, you could receive a message such as "The evidence were too accurate. Someone must have leaked them !"
4: How do you find who is that mole once you know that there is one ?
If the mole is in the group which has been arrested, the police could free it in front of everybody, or reward him. (Dirty conservatives ! They know, that will be beaten by all remaining liberals !)
If you have suspicions, you can Activate an agent to follow someone. You would gather some info on who he meets, and what he does during the day (this would require to define a kind of schedule with random events, and coordinate it with the different activities. Hmm. Maybe not, we just need to follow him when he is not working with you. So make out random events, such as he meets his sister...). That person would meet a number of other people, or drop a letter somewhere or give a call, drop a packet. You could try to follow that new person, try to see the number, take the packet, wait for someone to come... Then that contact could appear to be some normal contact, not to worry about, or go directly to the police (or give a call to the police). We could imagine, 1, 2 or 3 intermediates to make a real investigation.
Of course, there would be a check of intelligence, desguise... Any liberal would like to be followed (and thus might loose heart, juice...). Or the mole would stop which has seen you would stop giving the info (instead of seing his contact, he would do something else).
5: How to avoid to get one ?
Hmm. I don't think that you can avoid the check against getting a mole or not. But, if some moles have been caught and suppressed in an examplary way, that check would be more difficult. If your liberals know that they are being surveyed in the organisation (They know that you "A"ctivate some people to watch what is happening), they check would be more difficult too. So you exchange the juice reduction of those who have seen that they have been followed, with a bit more security. (Hey, a terrorist organisation which does not get paranoïd is not a real terrorist organisation !)
By the way, when you activate someone to follow someone else, you have to actively specify who he has to survey. I wonder whether the tracking described above should be automatic, or wether you should be given different choice. But, an interesting option could be "show yourself" : the liberal (whether a mole or not) would know that he is being followed, get the juice reduction but also increase in the felling of being surveyed (and if the guy is a mole, he would stop its action. You never know that he was one, but at least the info he gave has been stopped).
Oh, if there is a high feeling of being surveyed, each mole would be less active.
6: Okay. You have found one. How do you design it ?
In fact, you could designate anyone as a mole, and thus be wrong. Designate someone wrongly would have a very bad impact on your team or liberal agenda. (and probably good if you find a real one). Maybe the way to do it would be when following someone, you also have the option of "F"ight ! and "K"idnap, like any conservative. Then the bastard could be torturred. They idea is not to convert him, but let him suffer as much as possible (yuk !) and make an example, eventually get some info on what he told exactly (you might get an insight on who has high heat on him, whether one of you safe house is not safe anymore, or where he hid the stuff he stole. He might even give an other mole. Or his contact that you might follow. Maybe that man is in contact with other moles...)
Well. That's probably about it... What do you think of it ? Of course, any tiny bits here would need more work to be defined.
Hmm, yep, if we decide that it would be fun, it would probably take a bit of work to implement that ! Maybe I should help a bit... (but might development environment is broken right now)
Lets go play a bit now !