Use the Fiddler
Your outline is pretty solid. It doesn't sound like the lies your PC is telling are outlandish or particularly egregious, more that they lie constantly out of convenience. I don't think this will be a problem for you given the sort of campaign it looks like you are running.
Lying to this one NPC isn't a huge deal. You have a contingency plan already set up in the form of another faction the players can join. They lost their chance to meet up with Equilibrium, but that isn't hugely disruptive and the shady Fiddler faction sounds like it will fit at least your thief character a lot better. Your other PCs have the opportunity to contradict the thief or make time to talk to the Equilibrium agent on their own time, so if they pass that up it seems reasonable to have them not end up with that faction, at least for now.
You should move them to the boxing match, which I assume you set up for another PC's enjoyment in particular. The thief will lie the party out of or worse into the allegations of fight rigging (which might also, by that point, be true). Then the Bookie will offer the 1000 gp for the PCs to accompany him to an island, the players probably say yes cause there's not really any lying involved and he already knows who they are somehow it seems like, and they end up meeting the Fiddler.
The Fiddler seems to be a 'my sources tell me' type dude, so the only thing I see them lying about is their involvement in 'freeing gluttony', whatever that means. This faction seems like it is full of social manipulators, liars, and thieves, so your thief's lack of inhibition regarding convenient, plausible lying shouldn't be a problem here but rather an asset.
Lying in character can be a problem in the same way always killing everything that moves can be a problem. Lying is a weapon, and using it hurts people. While you are probably willing to tolerate a certain level of immorality from your players, actively suicidal and nonsensical play (e.g. lying to the town guard about the existence of a fire just to see them freak out, lying to the other PCs about the results of a scouting expedition despite the fact it will get everyone killed, lying to the guard to convince them the character is a murderer when really they're not, etc) seems like it would cause a lot of problems in your campaign. In truth, such play causes problems for most groups. You shouldn't allow that sort of behavior from a player on the grounds that it would ruin the game for everybody else. It doesn't sound like that's what's happening, though.
Reasonable lies (e.g. lying to the town guard about the non-existence of a fire to cover for the party while they put it out to avoid getting in trouble, lying to the guard to convince them they're not responsible for the recent death of a nobleman, etc) shouldn't be a problem for your style of play. Where that line of reasonability is is something you will have to figure out for yourself, but generally if the character makes sense and their lies make sense in terms of furthering their agenda (e.g. lying about seeing the bandits. "Don't give free information to strangers, especially potentially compromising information about yourself") things will probably be ok. Just expect them not to admit that they or the group are important, nor confirm nor deny the existence or details of any group activity without something in it for them.