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I doubt there is much market for such a thing so your desire for "reasonably priced" is probably a pipe dream. If you just want "the look", you could buy a stock canoe and just paint it yourself or hire somebody to give it a paint job. Or build your stitch and glue or woodstrip canoe from a kit and paint it up.
Watch your local Craigslist ads for old canoes for sale. Back in the 50's and 60's there seemed to be faux birchbark canoes around with funky paint jobs. I have come across old Michicraft boats with that sort of finish listed for sale. A vintage canvas on frame Old Town could be painted to look fairly authentic.
Are you looking for this just for display purposes or do you actually hope to paddle it, like for black powder or Mountain Man "authenticity" get-togethers?
Google is your friend. Try searching "reproduction birchbark canoes"