Can I fly a "Contact Approach" to this airport?
Here are the requirements for a contact approach from the AIM 5-4-25:


  Pilots operating in accordance with an IFR flight plan, provided they
  are clear of clouds and have at least 1 mile flight visibility and can
  reasonably expect to continue to the destination airport in those
  conditions, may request ATC authorization for a contact approach.


If all those things are true then you can request a contact approach (ATC can't offer it). But ATC has requirements to meet too:


  Controllers may authorize a contact approach provided:  
  
  
  The contact approach is specifically requested by the pilot. ATC cannot initiate this approach. EXAMPLE− Request contact approach.
  The reported ground visibility at the destination airport is at least 1 statute mile.
  The contact approach will be made to an airport having a standard or special instrument approach procedure.
  Approved separation is applied between aircraft so cleared and between these aircraft and other IFR or special VFR aircraft.
  


If those things are also true then ATC can authorize it but there's no guarantee that they actually will.

In the example you gave, it looks to me like all the requirements are indeed met (although you didn't say whether the two-mile visibility is flight, ground, or both), so a contact approach would be legal. There's no requirement to comply with the minimums for a published approach during a contact approach.