The ‘mouse-shaped deposit’ meaning is more properly linked to the word ‘souterkin’, meaning, per the OED
An imaginary kind of afterbirth formerly attributed to Dutch women
The earliest usage example in the OED is from 1658 and says:
There goes a Report of the Holland Women, that together with their Children, they are delivered of a Sooterkin, not unlike to a Rat, which some imagine to be the Off-spring of the Stoves.
And from 1862:
The housewives of Holland no longer bring forth sooterkins by sitting over the lighted chauffers.
The term, per the OED was also applied to people, specifically as a designation for the Dutch:
Ye Jacobites as sharp as Pins, Ye Mounsieurs, and ye Sooterkins, I'll teach you all the Dance.
But also applied to those of dark visage whether by occupation or nature:
The highwayman pushed poor Sooterkin [= chimney-sweep] out of the way.
Here is the sugar beside, which the hands of the sooterkin negro Reared [etc.].
Regarding the references to stoves and chauffers: in that period in the Netherlands women would have small ceramic stoves that they would tuck under their skirts when doing stationary work. Hence the idea that things under the skirts might get ‘sooty’.
This site explains and hosts paintings showcasing their use:
Foot stoves were used to warm the feet and were a common accessory in the Dutch household. The stoves were constructed with a wooden box that was ventilated on one side with holes or a slab at the top. Heat was conducted by burning charcoal that was placed in a ceramic or metal bowl (brazier) inside the stove. The feet were warmed by resting on top of the stove.
So, while the definition of ‘sootikin’ which abounds online is a slightly bastardised version of the original, it is possible that sootikin is being used in the transferred meaning of referring to the dark features of the cat.
The OED does offer another, obselete, and older definition, from the 1500s in Anglia, of ‘Sweetheart, mistress’ with an etymology of ‘older Dutch or Flemish *soetekijn < soet sweet.’
So possibly the use of ‘sootikin’ in the poem carries traces of this meaning also.