Your suggest of making part of the grid bike only is also a good one, but its more radical.
People will not even consider that unless there are already many bikers using these roads. You need to first have a bike lane and then you can make the argument it should be expanded.
> would need much less repaving work
Bike lanes shouldn't be paved at all. Its much better to use tiling methods, better for water and repeatability. You never basically need to repave them if cars don't destroy them.