Is it possible to print OSM’s in BaseCamp ‘exact’ on topo scale (e.g. 1/10000; 1/20000; 1/25000 …) and not on a variable scale of the printer?
No idea, I’ve never printed a map. I’d try to ask the Garmin support, the problem is probably not related to the source of the map data.
If you just need a topo of the area printed to scale, then CalTopo might work for you as it allows you to export PDFs at a scale you specify. Unfortunately they only seem to have contour lines for the United States so if you are outside the US you’ll have to find another site.
The maps at OpenTopoMap are quite nice too but I don’t see how to export or print an area at a specified scale.
I have a hack running at lab3.turepalsson.se/map which allows to-scale printing of topo maps of Sweden (with UTM zone 33 grids). It only has Sweden in its database and has no disk space for anything else, but the software could be set up to run with any database extract. The map style is trying to mimic the 50k topo maps which used to be (but no longer are) produced by the Swedish national mapping agency, which probably does not make much sense for more densely populated areas, especially since it involves lots of heuristics to associate OSM data with map symbols. (For example, it maps highway=primary to “public road, at least 7 meters wide”, which, I learned the hard way, does not work in the English countryside! 
) The software behind all this is currently undocumented and unreleased, but if there is interest, I could try to change that…