I have converted M12 tappings in blank arbors to 3/8 BSW by Loctiting in a piece of M12 studding and then drill and tap to 3/8 BSW.
M10 (0.393" )is 8.5 mm ( 0.3362) Tapping size (Core dia 8.159 mm 0.321" ) 1.5 mm (p 0.059" ) pitch
3/8 BSW is 0.3125" Tapping Size, (Core diameter 0.295" ) 16 tpi (0.06252 ) pitch
The sizes and pitches seem all a little too close for plugging and retapping to be feasible.
My method would be to make up a drawbar with a M10 thread on the end, if that will pass through the spindle.
Howard