For this question
Let's settle it _________ race.
Can we say let's settle it through a race? Is is it correct?
It sounds more natural to this US English speaker to say that we settle it with a race. For example,
"Settle it by _____" is also possible:
If any dispute arises out of this agreement the parties will attempt to settle it by negotiation.
but when using by we normally use a mass noun or a gerund, like "by ___ing" or "by means of" or "by negotiation"; using by with a count noun seems to be unusual, although I can't explain an exact reason why that is so.
So either "settle it with a race" or "settle it by racing" would sound fairly normal.