This was quite an interesting bit of research. These now-mostly-obsolete Japanese terms look very Latinate, but as many commenters have noted, the phonetics do not align as expected.
After some digging, I've figured out why -- they are not directly from any Latinate language. The source was actually English, much to my surprise.
The now-mostly-obsolete English source terms were imported into English from Old French via the Norman invasion of 1066, which explains the weird phonetic variances: English phonology has undergone a number of historical processes that did not affect the continental Latinate languages.
Between the Japanese Wikipedia article linked by the asker, and the limited uses of the English terms (just cards and dice), a couple puzzling points become clear:
- Why do these strange terms only cover the numbers 1 to 6?
- These terms were specialized uses for games using dice, and a single die only goes from 1 to 6.
- Why have these terms stuck around even though standard Japanese or standard English has other words for these numbers?
- Specialty contexts (like gambling) can create the conditions for a sub-culture to have its own jargon, which tends to persist as long as people are interested in that sub-culture.
- Once interest in that sub-culture wanes, the jargon terms themselves fade from use -- much as we see in both Japanese and English, where most of these terms have become obscure historical footnotes.
Etymologies back to Latin
Key: JA = Japanese, EN = English, ME = Middle English, OF = Old French, LA = Latin, * before a form = unattested in what I can find, † before a form = obsolete in modern mainstream English
- JA エース → EN ace → ME ās → OF as → LA as, originally meaning something like "a single penny"
- JA デュース → EN deuce → ME dewes, deus → OF deus → LA duo, "two"
- JA トレイ → EN †trey → ME treye, treie, treis → OF treis → LA tres, "three"
- JA ケイト → EN †cater, possible dialect form *cate (compare Norman French quate) → ME cater, *cate → OF catre, quatre → LA quattuor, "four"
- JA シンク → EN †cinq → ME *cinq, cink → OF cinq or cinc → LA quīnque, "five"
- JA サイス → EN †sice → ME sice, sis → OF sies or sis → LA sex, "six"
Note: I have not found any source that specifically says "these Japanese terms are from English", except for エース, which is quite well attested. Other than エース, these terms are entirely missing from my JA dictionaries. That said, English is the one language that has all of these terms, all with the expected phonology to match the katakana (with the apparent exception of cater, which should be cate to match the Japanese; while there is circumstantial evidence suggesting such a form might exist in English, I cannot find a solid source for this).