Can I get bathroom permit in old house (1950s) which doesn't quite meet modern codes?
The plumber most likely didn't bring it up because (1) he doesn't know much about the height requirements nor is he required to or (2) he is just there to do the plumbing or a combination of both 1 and 2. 

Next, basically the general rule is that things like this can be 'grandfathered' in, so long as they aren't touched. Since you were code compliant at the time of installation or construction (or somehow got by), they can not require you to upgrade anything to meet modern codes. However, if you ever go to do any new work or renovation like this, anything that is done must meet the current codes.

I did find, and this copy of the code is old (Seventh edition - 2009), an exception to the ceiling height.


  780 CMR: State Board of Building Regulations and Standards
  
  5301.1 - Exception 2:
  
  Ceilings in basements without habitable spaces may project to within six feet, eight inches of the finished floor; and beams, girders, ducts or other obstructions may project to within six feet, four inches of the finished floor.


This exception may no longer be around in the current code, and even with it you might not fit within it's parameters. 

(6'-8" overall from floor to ceiling, 6'-4" under the duct from floor to ceiling, but still 6'-8" regardless above the bathroom fixtures)

If you still do not meet this criteria (again, if and only if it's still in the code) then they do not have to permit the work and the job can not be done. I'd suggest talking to the town to see what can be done, they'll give the definitive answer.