I want to say
- peppermint ˈpɛpəmɪnt
What, if anything, determines whether I can do so, besides accent?
There are several factors that make /ˈpɛpəmənt/ an unlikely phonemic representation of "peppermint."
In general, only unstressed vowels can be reduced. In theories that distinguish multiple levels of stress in English words, we can narrow it down even further: only fully unstressed vowels can be reduced.
Actually, the main criterion for figuring out if a syllable is fully unstressed is often the presence of vowel reduction, so this is not very helpful in this case! There can sometimes be other indications, such as t-voicing in dialects that have that. (For example, the /t/ in "humanity" is generally voiced and flapped in American English, but the /t/ in "manatee" is voiceless. This is evidence that "manatee" has secondary stress on the last syllable). Unfortunately, the last syllable of "peppermint" does not start with a /t/.
Fortunately, there are some more general principles that we can state about secondary stress. One of them is that the second element of a compound word gets secondary stress in the same position as the primary stress in the independent word. There are a few exceptions (like words ending in -man /mən/) but the exceptions are rare and mainly confined to morphemes that occur as the second element of a great many compounds. Mint does not seem to fall in this category, so we would expect it to have secondary stress in the compound word peppermint, (ˈpɛpəˌmɪnt) which would prevent vowel reduction from taking place.
Even if someone pronounced "peppermint" without a secondary stress on the last syllable, it doesn't necessarily mean the vowel would be reduced to /ə/. In some analyses of some accents, /ɪ/ in an unstressed syllable is already considered to be a fully reduced vowel, that cannot be reduced further to /ə/. This is to account for the contrast between words like "Lenon" /ˈlɛnən/ and "Lenin" /ˈlɛnɪn/ in this kind of accent. People who do not distinguish unstressed /ɪ/ and /ə/ are said to have the "abbot-rabbit merger" or "weak vowel merger."
That said, it's a common poetic license to move the position of the stress in a word; I see no reason why this wouldn't also apply to removing a secondary stress. So if you need to use the pronunciation /ˈpɛpəmənt/ to rhyme with some other word ending in /ɛpəmənt/ (sepiment?) I don't think it would raise any eyebrows.