Not sure to what level the OS maps are routable, but the ‘Openfietsmap Lite’ map is very good for cycle routes / bridleways / paths (it’s based on Open Streetmap).
You can download it from the link above and install on Basecamp (you could also install it on the Garmin if you wish).
Basecamp still makes the occasional bizarre choice, but you can ‘undo’ and try clicking closer to where you want to go, and it usually gets the message. Switching routing profiles while plotting a route doesn’t seem to work properly, but you can force it into ‘direct’ mode by holding down one of the shift/control/alt keys to ‘bridge’ across a section that it thinks is un-routable (can’t remember which key, and it’s probably different on the Mac, anyway).
There are options within the Basecamp routing profiles which influence the choice of route which may be worth playing with.
Now the important bit:
When you get the route the way you want it, TURN IT INTO A TRACK to send to the Garmin. If you don’t, the Garmin itself will have a crack at picking a route through the points you’ve chosen. If the maps or routing preferences are different (at all) chances are the route gets mangled.
If you use a track, the path over the ground is preserved, and you can happily change maps.