For straight line speed there will be little difference between a road and properly set up touring bike...
Items that might need changing or that could slow you down:
-Install narrower tires than what you use for riding to the grocery store in Tibet (good quality 700 X 23 to 700 X 28, depending on your weight and what you like)
-Take off the racks, bags, and at least 4 of your 5 water bottle cages
-Make sure you have gearing so you will not run out of gears during the event
-If it is super hilly and you are as fit and lighteweight as you can be, a 7 pound touring frameset will put you at a disadvantage to an identical rider on a 4 pound frameset
-you will fall to the back during the first tight corner and slip further back with every subsequent corner
Most of these do not apply to you and a normal time trial. The reality is, amateur racers are seldom anywhere near reaching the performace limit of their bikes. If Alberto Contador or Carlos Sastre entered a local race on a touring bike he would still kick arse.
My advice: Run whatever bike you think you are fastest on and do your best.
$40 is a lot for one race, though. Maybe if you joined a club you could take part in their weekly time trials to track your progress over a season.
Also, if you join a club you generally make a lot of contacts with people who are selling or know others who are selling road bikes - if you want to be a little more competitive.