Never having done a criterium, take all this with large grain of salt.  I assume you will be in the lowest category.  Race with whatever you want or have.  I don't know if there are regulations concerning handle bars since this one has straight bars.  I do know you can't race with aero bars for instance, safety issue.  Check with race director before hand.  Will they think you are outclassed, probably?  Will some of them even laugh at you, maybe?  Sounds like that might be good motivation to put them in their place.  Treat it as a learning experience and have fun.  Don't crash, that's not fun.  Strip off all unecessary weight, see if you could borrow a good set of rims, put light racing tires on, that will help.  Good luck.  Post a follow up please.

Well,  to quote from an old Little Rascals flick..." WELL, WELL, WELL, you fell in the well."  You asked and have not posted a subsequent report so I presume the race came and went.  Aero bars can't race but straight bars on an FX bike are legal.  In a criterium race on a .8 mile course I've averaged, not top end, averaged 26 mph or more.  I'd say that aerodynamics does play a role at that kind of speed.  The trick is holding a strong pace through the turns and being skilled enough to do so.  Do you even have step in pedals?  The tires on FX bikes are wider and therefore heavier and therefore slower.  The advice to swap to a racing tire is good or even a race wheel too.  BUT your bike won't b equipped to keep up with the racing bikes out there.  That said, a strong rider could do it and kick butt as well.  You would be doing a Category 5 race or citizens race if they have it.  You will have to buy a one day license which is from the USCF.   To get back to your question, the same rider on a an FX bike will, simply put, lose.  A strong rider/racer would be able to sit in. Lance will win. 

My advice:  Start doing training rides with riders on road bikes and learn how to ride a paceline, sit on a wheel, learn the terminology of group riding, get step in pedals which you can put onto a new racing bike should you want to, and don't race. Save your money.  Go watch the races and talk to the racers.  Watch them in the turns, with their inside leg up, and note the difference in power from a cat 5 to a cat 1,2 race.  Cat 5's still rip.   Let us know what the deal is and if you want to start racing get good advice on how to start and what to do about the bike.  This forum works for that or click on my profile and email me. 

By the way, I love the FX bikes and would love to have one.  You can even get a $2,000.00 full Carbon FX bike.

EJ