Bikes are lumped under the same category as a horse where byways and bridleways or Rights of Way in general are concerned. #mybikeisnotahorse
Though that’s normally an annoyance in that we’re made to ride on paths unsuitable for bikes or destroyed by horses, yet banned from those that are ideal for bikes and suffer less damage riding on them (wide well bedded in footpaths), and even less damage sometimes than from walkers.
In this case though… I think the bike = horse still applies and sounds like okay so long as you’re not going to stop for camping.
Any purpose – depends if that includes passing through.
A byway that excludes vehicles including horses seems to be a reclassification of the RoW to not be a byway any more. Sounds more like a footpath.