The issue is that the name on your airline ticket is supposed to exactly match what appears on your ID.
All airlines were required to have their systems set up to take middle names by sometime last August, IIRC. So, I assume you booked your tickets before that or that your airline was out of compliance. (I don't think every airline had their systems updated by the deadline.)
If that's the case, TSA should be "flexible" as their official policy states. Of course, there are plenty of TSA agents out there who don't know understand their policies completely or just choose to be a pain and cause you problems.
If the airline will update your tickets to include all of your middle names, then you might want to do it just to avoid the (relatively small) potential that it could cause some delays at the airport. You all likely will be able to fly regardless as you can even fly without proper ID in most cases, but will have to go through extra screening.