Looking at your map link I suspect the Bridleway only ever went as far as the ( now ) disused quarry and that the correct ( ? ) route for horses ( and as such cycles) should be on the permissive bridleway which starts approx 300 - 500 m further down Keighly Road, this also links up with the marked cycle routes. ( zoom in by 1 level for a more detailed map )
As far as being allowed to push your cycle along a footpath, I understand that this is now up for debate after a recent court hearing , although in the past I think that the understanding was that it was not allowed as a cycle is not classed as a "natural " accompiament to perambulation ( labrat is the guy who would know the wording but he isn't around anymore )
TBH... If the new landowner is such a PITA , then I'd be inclined to use the Permissive route anyway as it negates the need to walk and by using it you are ensuring the evidence of use is maintained,.
For fun ( when in a devilish mood) I'd be tempted on occasion to just ride up and down the bridleway to the quarry, just to see how often the guy kept coming out to stop you .....