Great idea about them picking up a recorder! (I just feel Spock would have said "Telemetry suggest" rather than "Lab theorizes", then...)
But the outposts wouldn't suffer from those limitations you worry about in a model where the enemy can warp in under cloak and the recorders never visually witnessed a warp-in. They would do perfectly fine against visible ships, at warp or impulse. So well, perhaps, that it would be deeply indoctrinated into the base commanders that no revealing transmissions be sent under any circumstances short of the Last Trump. Any conventional threat could be assessed before it came into weapons range, and a choice made on whether to tell Earth that the enemy has finally attacked in such a force as to overwhelm the bases soon, or to rely on stealth, armor and armaments and deal with the enemy without compromising that first aspect of protection.
And yes, stopping for a while didn't appear to be a problem: the heroes were confident they could follow the invisible ship anyway. All the invisibility seemed to be good for was for fouling fine targeting; the general movements of the ship could still be followed with various sensors.
Up until that comet encounter, that is. But the "We grow visible" scene suggests that invisibility comes in degrees, and is dependent on power allocation. Perhaps the ability of the enemy to track ship movements would be acceptable for the Romulans when the only known enemy in the vicinity had just been reduced to dust, but a dial would be turned when a new opponent was detected and the ship would grow more invisible.