I didn't make it to the end of the thread, so I apologise if someone's already suggested this or if my suggestions have already been rendered impossible, but here are my thoughts"
A cold method: JE starts off with a name.. it could go anywhere at this point -- if he gets a Joshua, then fine, maybe he goes straight into a routine with that person's relatives. In this case, perhaps a combination of (a) the look of affirmation/recognition from Liam at Remie's table, (b) the look of bewilderment from other table members who have met Liam under the name "Liam" once he stands up [which seems to just be Remie.. did you react much, Remie? Did other people at the table react to Liam standing up?], and (c) JE's cold reading skills lead him to quickly notice someone semi-reluctantly acknowledging their name to be Joshua and standing up leaving their table partners looking bewildered. It is then pretty likely that Joshua didn't introduce himself that way to them. The most probable explanation for *that*, by far, is that they go by their middle name.
This theory relies on JE being pretty good at what he does, and probably seems easier in hindsight than it would be on the spot, but hey.. that's his job.
A hot method: Before and during the show, JE has people running plate numbers from vehicles in the carpark and cross-referencing details with some other info databases. I don't know how easy this is for non-LEOs in the states (I assume this is USA?). If you find that a car is registered to Joshua Liam Surname, and then you search some other (more open) national database and find a record with "Preferred Name: Liam", you've got a pretty sweet hot read to perform later.
This theory relies on some possibly not-entirely-legal behaviour if it's about running plate numbers.. but there are probably legal alternatives, depending on where else Liam has shared his personal information, what stickers he might have on his car, etc. etc.
Disclaimer: I don't know what I'm talking about and I've skipped most of the thread.