Kawaishi came from Waseda U. and the kata he taught reflect that background. Judo ranks, back then, were issued by the Kodokan or the Dai Nippon Butokukai, which was a sort of National Governing Body for martial arts. The Judo section of DNBK was headed by Kano's top guys (like Nagaoka etc.)
There's no reason to believe Kawaishi studied anything other than Kodokan Judo. I'd be interested to see any contrary evidence.
Remember that the terms "Judo" and "Jujutsu" tended to be used interchangeably in the early days, even by Kodokan practitioners.
The techniques shown in "Ma Methode de Self Defense" are pretty orthodox Kodokan Goshinwaza, mixed with a bit of Savate, a soupcon of Karate, and perhaps a bit of influence from Feldenkrais.
If anybody thinks it's koryu, please specify which koryu, and when Kawaishi would have studied it, and with whom.
Just for the record, my sensei was a pupil of Kawaishi in Paris after the war, and received his Judo shodan from him around 1950. If you look at the techniques in "Ma Methode de Judo," "Ma Methode de Self Defense," and Feldenkrais' two books "Judo" and "Higher Judo" you'll have a good idea of the Judo curriculum I grew up with.
Yours in Judo,
Brian P. Griffin