The overriding concern of the constitution is that the federal government not tromp all over the rights of the States. It would not be reasonable for the constitution to have provisions to secure the right of the federal government to arm its police, but it does seem reasonable to me to have provisions to ensure that the federal government does not interfere with the right of the several States to arm their police.

However, multiple states specify in their constitutions who comprises their militias. In Virginia, for example, it is "composed of the body of the people." In Illinois, "The State militia consists of all able-bodied persons residing in the State except those exempted by law." Given such constitutional provisions, it seems unreasonable to think that police were in view when speaking of a "well-regulated militia."