Linguistically speaking...

To me, to say "Watazumi Doso" is like to say he is my great teacher. Doso (道祖) means founder of a school or originator of a practice (way). Equivalent notions may be shuso (宗祖, or founder of a sect), soke　(宗家, or founder of a school), and iemoto (家元, or head of a school). But we don't call "Kurosawa Doso" or "Nakao Shuso" (a family name followed by "doso," "shuso," or whatever). It's not impossible to say "Kinko-ryu shuso" or "Tozan-ryu soke", but we normally don't say like that because these two schools are too large and have many subgroups like other hogaku groups (e.g. Miyagi schools). We surely say "Chikuho-ryu soke" or "Seien-ryu iemoto" as these schools are small enough to form groups (ryu or school is a bounded system).

I often say "Watazumi Fumon" instead of calling him "Watazumi Doso." (Of course, this doesn't mean I don't respect him). Before he became famous, he introduced himself as Tanaka Fumon. So it won't be so strange to call "Fumon ryu," like "Kinko-ryu" or "Tozan-ryu" (a first name followed by "ryu"), instead of "Watazumi-do."

Whether Yokoyama's style should be called Watazumi-do is another question. It's beyond the matter of language. Some people carefully avoid using the term "Watazumi-do." I personally enjoy calling it "Wadatsumi-do Yokoyama-ha" (modeled after "Myoan-ryu Taizan-ha"). Just a language game.