People fly pregnant all the time. Flight attendants and pilots usually fly part of their pregnancy. One xray equals a few days of nornal radiation packed into one day. You probably have been in areas of higher and lower radiation anyway during your pregnancy. Someone’s house you were in had a lot of radon or their granite topped kitchen with marble floors was in a small space and had more radiation than your countertop and wood floor at home. Or, you laid out at the pool off and on for a couple weeks this summer and got a tan. If I were you I would not do the xray at security and opt to be pat down.
You definitely should mention you want to fly to your doctor and make sure he thinks it will be safe for you. If he has any concerns of early labor he might advise against it.
I disagree with @AshlynM about the morning sickness. If you don’t have morning sickness from the hormonal changes there is no reason to think you will be sick on a flight.