Most airlines specify that only ‘sports equipment’ may be carried in ‘sports equipment luggage’. So helmets and SPD shoes are fine in a bike bag. I’ve carried all sorts in bike bags and never had any issues. Remember that it’s the airport authority, not the airline, that scan the bags, and they aren’t really interested in the contents as long as they’re not bombs and guns. The only issue they care about is the weight, so make sure you have sufficient weight allowance for your trip. With Easyjet, it’s worth buying the ‘sports equipment’ allowance as well as the hold baggage premium, as that then gives you a total of 32kg for the bike bag. Other airlines might just let you have a total weight allowance with no other conditions.
Get it sorted before you fly though, as it can be a right hassle at the airport.
Top tip: if travelling with a bike in a bag, go round the side at the check ins rather than queue up. Feign ignorance/stupidity, and the staff will invariably let you check in at the premium/business/fast check in desk just to get you out of the way. I have never queued at an airport whenever I’ve travelled with a bike.