You’re never going to better the price of taking your bike on the plane with you – although check on multiples as that might then be ‘excess’ excess at a different rate.
From Sheffield to LGW, you can do a 1 stop train in St Pancras… not a bad station to deal with.. then on the slow train out to LGW.
Or, 6+hrs on the National Express bus, but thats an awful journey.
If you’ve not booked your flights yet, why not look at Manchester Flights with Canadian Affair/Transat.. ask a mate to drive you there.
Boxed bikes more of a hassle than wheeled bags, but we’ve seen plenty of diy solutions to get boxes rolling!