Flybe’s rules on bikes are:
Bicycles
We accept the carriage of bicycles on a stand-by basis only. Carriage will thus only take place if there is sufficient space available in the hold and is subject to a fee. To ensure the safe carriage of your bicycle, we require that your bicycle be prepared as follows:
Handlebars must be turned and locked in line with the frame.
Pedals must be removed or turned inwards.
The front wheel must be removed and securely attached to the frame.
The tyres must be fully deflated.
The bicycle must be bagged or boxed, wherever possible
It seems you can take any size hold luggage as long as it’s under 20Kgs as you say.
I don’t see what risk you take just turning up with your sub 20Kgs bag and just checking it in, in fact you take less risk as they only take bike as standby only, so it could be left behind and you don’t seem to be able to pre-book them.
They used to be the best airline for bikes – they flew under BA’s old rules which meant bikes were free! Sadly, it’s not the case any more for Flybe or BA.