Personal opinion only:
I find that a sharp but noisy picture is usually preferable to a less noisy out of focus picture. Exceptions may be in situations where artistic aspects make the achieved degree of blur acceptable. Your findings may differ.
(1) Lowest speed - SSL : Shoot at as low a speed as you can and still tolerate the target's & your motion artefacts - largely motion blur - BUT NOT SLOWER.
(2) Fastest speed - SSH : Shoot at the highest speed that you can that allows noise and quality to be acceptable.
If SSL <= SSH then you have a ball game.
If SSL > SSH you have a problem.
Solution - any mix of:
A camera with better low light performance
A faster lens (or larger aperture if available)
'Better' post processing software
Add light to the scene
Improve your technique
Why has nobody so far at least mentioned Flash? Not always what you want and tends to change the 'environment' but has it's place. I was very surprised some years ago to see an International competition (relating to wild life afair) won by Spanish photo of a wild wolf jumping a low fence at night. Remote auto trigger used. They have details of film speed, camera type and many settings and more. AFAIR ISO 25 was used ! NO mention was made of the fact that flash must have been used. If you can do that and win such a prize then flash for the black cat may just be OK :-)
As others note, overexposing does not sound wise here.
You are effectively decreasing shutter speed all else being held equal.
Sometimes blur can be acceptable, or somewhat acceptable. Here the result was accidental - shutter speed was 1/6s & flash was used. Without flash this is a reject. With flash it may be acceptable depending on other shots.
FWIW: NIKON D700, 1/6 seconds, f1/6.30, aperture priority, ISO 3200, 260.00 mm (genuine as FF Camera)
AND this is what caused that camera behaviour: Flash Setting - SLOW
Not a setting I usually use. Camera ensures that background light 'rounds out' any underexposure (as opposed to metering scene and then hoping)(if I understand slow mode correctly).
Pushed to limit (or beyond :-) ).
This was purposeful (believe it or not. )
Camera set to rear sync and several seconds exposure and swung in arc around background and then stopped at selfie position just before rear-synced flash flashes. Results variable. Whether any are "any good" is in the eye of the beholder :-). (I like SOME of them). Tracking black cat down tree may be interesting :-).