The "San" should, indeed, rhyme with "can", at least in the usual English pronunciation.
The "soo-yee" or "soo-ee" part is slightly different according to which language/dialect you are speaking. Since it is based on the (originally Chinese) characters for "mountain" and "water" (i.e., sea), it is read as a "local" word in China and Korea, as well as Japan, since those countries also use Chinese characters (although only partly and more historically in Korea).
Assuming you want Japanese, because the brand comes from Japan, then the above transliteration comes pretty close.
In Cantonese (the Chinese dialect spoken here in Hong Kong) the pronunciation of "San" rhymes more with "Kahn" and the second syllable is closer to "soy" (although it is subtlely different in ways I can say, but have no idea how to reproduce in English writing).
In Mandarin, the pronunciation would be more like "Shan" + "shway".
Oh, and the name is said (or at least thought of) as basically TWO syllables in any of these languages/dialects, NOT three syllables as many English speakers might want to do with the "san-soo-ee" breakdown. The last two sounds blend one into the other as a diphthong (like "oi/oy" in English: you don't say "boy" as two syllables just because the sound is written "bo-yee").