The first row of jacks and the one on the left on the second row are from sheets and decks which are in the collection of the French Bibliothèque Nationale de France. Only the middle one in the top row comes from a sheet which mentions the name of the publisher: LeFer from Paris. All the other decks don't give the name of the publisher. However, the Bibliothèque Nationale also has 2 uncut sheets, which have the same figures but in a different order. So there must have been at least two different plates to print the courts from. It's unknown who printed the sheets, but the different versions of colouring and the adding of pips and sometimes indices suggest that each publisher could make their own version, even a monotone one.

© Bibliothèque Nationale de France

© Bibliothèque Nationale de France

© Bibliothèque Nationale de France

© Bibliothèque Nationale de France

© British Museum

© J.J. Muller :)

All the jacks on the top row are pure single imaged cards. However, at some point a publisher decided to add pips and ranking dots at the bottom too, making the courts still distinguishable when held upside down in your hand. It could be seen as a first sign of the need for a double imaged pattern. But players had to wait until after Gatteaux's single imaged new standard French pattern from 1816 was replaced by a double imaged one in 1827. Note that the David design here below already has a small pip and index at the bottom.
The first jack here is from an unknown French publisher, but his placing of the dots interferes with the named courts. The published decks by J.T. Dubois show a better solution. The second jack is from a deck at the British Museum, with the same Dubois stamp as my deck. It seems that of all the different publishers only in the J.T. Dubois decks the dog is overexcited to see their boss. Those naughty Belgians!
However, in a later edition by Dubois (see below) the dog had lost his excitement again.

WWPCM

© Bibliothèque Nationale de France

LEAVES THAT QUESTION:
Was Gatteaux the first to introduce this dog?

ANWSER:
Yes... and no!

On the left is the jack of spades as designed by David and on his shield you'll find the image of our dog. It's there for no apparent reason, but Gatteaux took this image and gave it a more prominent place to illustrate the bond between a hunter and his dog. Maybe the dog even gave him the idea to create a hunter instead of a soldier. You never know how the creative process works within an artist's brain.