I disagree that those two arguments can be made effectively in parallel. Perhaps in an ideal world they could be, however in practice, one argument is offering those who would seek to diminish our rights a concession of sorts. By arguing the police who are a subset of civilians should not be allowed certain types of weapons opens the door to the reasoning that no one, or all civilians, should be allowed those types of weapons.
Additionally by allowing a further categorization based on military or non-military use creates more issues based on current and future issued weapons. If we concede that because select fire weapons are used by the military no civilians should have them regardless of LE status, how do we prevent the proverbial camel from coming into the tent based on that. M9s aka beretta 92s are military weapons, m24s aka remington 700s are military weapons etc.
I think the much stronger argument rests in the second line of thought, that if police are allowed these weapons we should be too. While it may not be effective in repealing the NFA or the Hughes amendment(as realistically I do not believe this to be achievable in the near future), it at least staunches the flows of anti-rhetoric which has broadened the target from actual select fire weapons, to those that are similar in design but semi-automatic. By proffering the argument ourselves that police do not need certain weapons, we risk those who oppose general firearms ownership running with that statement to the exclusion of other arguments. In short, I see it as a dangerous gambit to argue that LE should not have something, while attempting to keep the gap between LE/Mil and Regular Joe/Jane, to a minimum, and while attempting to preserve and regain what bits of the 2A we have left.