I'm curious if there is any difference in nuance between these two sentences:
I'm aware that grammatically speaking both are 100% right, but this question is not targeted at this issue.
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Your sentence 1 is ambiguous with respect to the scope of
'He washes his car at least once a week.'
'He at least washes his car one a week. (He also changes the motor oil once a month.)'
The first meaning is the same as your sentence 2, but the second meaning cannot be expressed by sentence 2.
In general, the position of an adverb often gives different possibility for its scope.
I don't detect any difference in meaning, but splitting frequency expressions (週に一度, 年に３回, etc) as you did in the second sentence sounds disjointed. As far as general adverbs go, technically you can put them just about anywhere, but most often you'll find them right in front of the verb they modify.