What are "approved weather reporting services" and do they apply to part 91?
First, within the context of the question, the term "operator" in the Alternate Mins section refers to a person or company that has an operating certificate.  Not just Joe the Pilot who is "operating" an aircraft. So, a "Part 91 pilot" (taken from the context to mean: "pilot flying for pleasure or personal business") cannot file this airport as an alternate when the tower is closed.

Second, the term "approved weather source":  In the FARs, various Parts apply to various operators (121, 125, 135). They each have similar verbiage about approved weather reporting services.  For example, Part 121.101 (for domestic and flag operations) says the following (paraphrased):


  No certificate holder may use any weather report to control flight
  unless it was prepared by the U.S. National Weather Service or a
  source approved by the U.S. National Weather Service. Each certificate
  holder shall use an approved system for obtaining forecasts and
  reports on each route to be flown and at each airport to be used.


You can see this related question: What sources of weather are approved by the U.S. National Weather Service? for more information.

BUT... the important point from that FAR Part 121 quote is the second sentence. ALL operators must have Operational Specification (OpSpecs) approved by the FAA before they can operate. One of the items within the OpSpecs will specifically address how and from where these weather reports and forecasts will be obtained. 

So, this is the bottom line answer to the question: only "operators" can use this airport as a filed alternate when the tower is closed, since that operator will have their own "weather source" that is approved by the FAA and listed in that operator's OpSpecs.