Can a pilot over the age of 65 exercise the privileges of a commercial pilot license under FAA Part 91 in the Bahamas?
There should not be a problem as long as you are flying under Part 91 because the age restriction which you refer to does not apply to all commercial pilot operations.

14 CFR 61.3 is consistent with the ICAO rule which you refer to and covers a US pilot, flying a US registered aircraft internationally and states:


  (j) Age limitation for certain operations
  
  (1) Age limitation. No person who holds a pilot certificate issued
  under this part may serve as a pilot on a civil airplane of U.S.
  registry in the following operations if the person has reached his or
  her 60th birthday or, in the case of operations with more than one
  pilot, his or her 65th birthday:
  
  (i) Scheduled international air services carrying passengers in
  turbojet-powered airplanes;
  
  (ii) Scheduled international air services carrying passengers in
  airplanes having a passenger-seat configuration of more than nine
  passenger seats, excluding each crewmember seat;
  
  (iii) Nonscheduled international air transportation for compensation
  or hire in airplanes having a passenger-seat configuration of more
  than 30 passenger seats, excluding each crewmember seat; or
  
  (iv) Scheduled international air services, or nonscheduled
  international air transportation for compensation or hire, in
  airplanes having a payload capacity of more than 7,500 pounds.
  
  (2) Definitions.
  
  (i) "International air service," as used in this paragraph (j), means
  scheduled air service performed in airplanes for the public transport
  of passengers, mail, or cargo, in which the service passes through the
  airspace over the territory of more than one country.
  
  (ii) "International air transportation," as used in this paragraph
  (j), means air transportation performed in airplanes for the public
  transport of passengers, mail, or cargo, in which the service passes
  through the airspace over the territory of more than one country.


Each of the bolded sections (added by me) show how each part is ruled out for a Part 91 operation as you describe.

Along the same lines, the ICAO SARP that you are referring to governs "commercial air transport operations," and not all commercial operations.  Part 91 flights do not fall under the annex with these requirements.

That being said, it is a foreign country and they can make their own laws which do not necessarily have to agree with any of this.  To be sure, you can always contact the FBO that you intend to use and ask them, or obtain a copy of the Bahamas AIP (I wasn't able to find a free copy) and see what it says.

Being based in South Florida and doing lots of flying to the Bahamas, I haven't heard of a problem with this (although I haven't specifically been looking for it either).