On the contrary, [u] is much closer to [ʊ] than [a] is. [ʊ] is even an allophone of /u/ in Russian (if Wikipedia is to be believed, at any rate: I don't speak Russian).
The only available options are бут, бот, бат—boot, bot, butt, sort of, but really the vowel sounds aren't quite the same. Buttchair is probably as close to a Russian pronunciation as Botcher, which I realize isn't an improvement on the funny-homonym front.
In my mind's ear, I'm pretty confident a speaker with a Russian accent, who hears the English word and repeats it, is accenting it towards бу. So I'd use that.
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(boot, baut, bot, I'd approximate them.)