What power do spirits have that resurrected beings do not? According to Mormon doctrine, Resurrected beings have physical bodies that can only be in one place at one time, which is why the Holy Ghost remains a spirit, so that he can dwell in the hearts of men:


  "The Father has a body of flesh and bones as tangible as man’s; the Son
  also; but the Holy Ghost has not a body of flesh and bones, but is a
  personage of Spirit. Were it not so, the Holy Ghost could not dwell in
  us." (D&amp;C 130:22, emphasis added)


It appears to me that some form of power or ability is lost at the resurrection. An essential power that only spirits have, which resurrected beings do not, hence the necessity of the Holy Ghost remaining a spirit.

It seems somewhat paradoxical to me that we would have anything to lose at the resurrection–if anyone understands what I'm trying to communicate by phrasing it that way... Obviously the Holy Ghost is going to be resurrected at some point (I feel it's safe to assume so at least), at which point in time he will forfeit the power that makes his role in the Godhead essential, and receive a resurrected body. My question is, what exactly is this power that the Father and Christ cannot wield as resurrected beings? Or where do I err in my pondering or phrasing of my question?
What an excellent question, as it forced me to really think it through.

I think the disconnect is in the very first lines.  You make two assumptions that Mormons would not agree with.  The first is that without a physical body a spirit is able to be in more than one place at a time. 


  Doctrine and Covenants 131:7-8
   7 There is no such thing as immaterial matter. All spirit is matter,
  but it is more fine or pure, and can only be discerned by purer eyes; 
  
  8 We cannot see it; but when our bodies are purified we shall see that
  it is all matter.


the second assumption you made with which Mormons would disagree would be to lump the Holy Ghost in with 'Spirits'.  We do have doctrine that states that He is a personage of Spirit without a body of flesh and bone but we know very little about Him.  We don't know if He is yet to be born into mortality, already passed through a mortal state, or never to be born at all.  We don't know how He accomplishes His divine mission but we don't assume that because the Holy Ghost is capable of it, then all spirits must be equally endowed.

What we do know (what we have doctrines declaring) is that we can become joint heirs with Christ, receiving all that the Father hath.  We can become like Him and part of the process includes being resurrected into a glorified physical body.  We saw an example of such a body in Jesus Christ and we saw that it (His glorified body) was capable of things which ours are not (see John 20:19). 

There is no regression in resurrection.