Just used Paisley freight freight for the first time and apart from the APC driver being a bit surprised by the size of the box it seemed to work ok.
Collected on Tuesday and delivered on Sunday, not sure what it was doing between Wednesday morning and Saturday morning though.
I had always used Parcel Force before as I could drop off at the depot. I did try this time but wouldn’t accept the booking as you found.
TNT and FedEx ship bikes but not sure if an individual can use that service. Have you tried something like Parcel Monkey that searches through providers, not sure if even that will let you select a big enough box.