Bridleways are primarily for horse riders AND pedestrians, not just horse riders.

Just because some bridleways may be unsuitable for bicycles doesn't mean that it should be avoided as a way type. If you wish to convey that a way is suitability surfaced, consider adding
https://wiki.openstreetmap.org/wiki/Key:smoothness
https://wiki.openstreetmap.org/wiki/Key:tracktype
tags to the way.

Changing tags just so the path appears blue on a specific renderer is 'tagging for the renderer' which is frowned upon. Other cycling specific renderers such as CyclOSM already render it as blue based on the surface type.