Hi Maryn,
Due to common use, I'd vote 10K. CMOS doesn't have guidance on this specific issue (as I found to my chagrin a year or two ago when I tried to look it up). The closest thing I found was CMOS 9.16, which covers numbers with distances such as "50 km (kilometers); a 50 km race," but that doesn't quite apply to what you're asking, since in most contexts, race distances are referred to with K (or k), not km. Still, that guideline is enough to land me solidly on the numbers side of things. From there, I'd apply the general guideline of following the more common presentation; in this case, 10K versus 10k.
CMOS's general rule is to write out numbers in dialogue, but there's that whole "This practice requires editorial discretion" bit, and given their examples of when to opt for numerals, I'd go with 10K in dialogue as well.
However, seeing 10K at the beginning of a sentence would make my inner editor cringe (at least in most contexts). Is it possible to reword those instances so that you don't have to start a sentence with a number? If not, I think I'd recommend sticking with 10K there (much like military time remains in numerals even if it starts a sentence). But rephrasing would be my preferred option.