I thought TMP universalized the arrowhead badge, but the screencaps aren't backing me up.
They are not?
Admiral Kirk of Starfleet (not Captain Kirk of the USS Enterprise)
is wearing the delta, thus you are right.
The one thing that stands in the way that the insignias presumably refer to assigned sectors of fleets would be "The Doomsday-Machine", IMHO.
Right from the beginning Spock states that they had charted the system in the previous year, so what's the Constellation's
business in their "backyard", as it obviously has an insignia quite different from the one of the Enterprise