Firstly is your race pool based or open water??
Have you hit the cold remedies to try & shift it?
Are you sure it is a cold? There are some people (myself included) who have a sensitivity to the chemicals they put in pools, I frequently end up having sneezing fits and a blocked nose (feels like a head cold) if I don't take antihistamines before I swim! So if you've swum recently this might be the problem!
In theory I don't believe there are rules against doing the backstroke, unless the race referee has specifically stated in the race information that it is not allowed. Check your race information pack! But bear in mind that to attract attention if you need assitance during the swim you should float on your back with an arm in the air!! If in doubt ask at the race breifing!
If your swim is in a pool, you could always do the breaststroke in the style of a little old lady, i.e. by keeping your head out of the water. It will slow you down but it might help!
Good luck and enjoy yourself!! 