I'd use an "eye dropper", a syringe, a straw, etc.. to transfer drops of solvent to the brushHow about spraying the brush with something like brake cleaner between dips in the bottle?
Or using a thin patch with the brush to keep it clean.
I do all the time, but only in the smaller bottle of Hoppes that I refill from a much larger bottle. I just rinse the small bottle out before I refill it.