That's 172's for you! They can be very twitchy if you are too fast, or have any x-wind. It really should have been a go around after the first bounce. I can't see why it would not be an option. It is a classic Cessna accident to wipe the nose wheel off on the third or fourth bounce. Seen it happen a couple of times, and it is not pretty!
Not sure I buy the 'thermal' theory though. More likely to be pilot induced, especially at the stage of your training. You did the right thing in not pushing forward after the bounce/float however. That invariably would lead to a loss of nose wheel.
Chin up and get back into it straight away. It's only a minor issue and practise is the only thing to cure it.
One little tip, when approaching the flare, make sure you are looking at the horizon, physically lift your head up, it allows your peripheral vision to work and it is much easier to judge vertical rates of descent.
Good luck!