I believe if you're a US citizen vacationing long-term in France because your work prospects aren't great at the moment - you'll still show up as not in the labor force in the labor force participation rate: https://fred.stlouisfed.org/series/CIVPART
But you won't show up in u-1 through u-6.
> Discouraged workers (U-4, U-5, and U-6 measures) are persons who are not in the labor force, want and are available for work, and had looked for a job sometime in the prior 12 months
And, as you say, there are labor force participation numbers if that's what you're interested in.