No, the 2210 is the more bike-friendly. However, there don't seem to be any advance tickets available on that, so an off-peak single is £15.40 to use that train. On the 2202, there are advance singles available for £7.
I'm happy on either. If you choose the 2202, bring bungees or some other means of securing your bike so that it doesn't swing around while hanging from the ceiling hook. On the 2210, all you'll need is a means of holding the brake on, as your bike's wheels will be on the floor of the train.
I'll TRY not to bore you too much about trains...