They certainly did make a range of their own air rifles and pistols, but I can't be sure that the Gem types they sold weren't German or Belgian imports, as (as far as I know) no examples have ever turned up. Mind you, they were not great exporters so all their guns are very rare, and only likely to be found in France. According to their catalogue description, if they did turn up they might have been marked "La Guillaume Tell" or even more confusingly just "Tell".
Only relatively recently did this pistol came to light, which is one of theirs, called the "Scolaire No. 2". It is a copy (improved) of the EG Dolla-type pistol of the period.
Another catalogue picture of one of their pistols shows it to be very similar to, but not identical with, the Jean Marck Gem air pistol. One of these has never been found, but I live in hopes.
Sorry, I have never come across the name 'Coire' in connection with an airgun.
Cheers,
John