I regularly ride a 1-gear bike with road brake levers on its mustache bar. After I saw this thread, just for ****zngigglz while riding it I tried to imagine brifter shifting on it and to be candid it didn't feel like anything I'd be interested in.
I think the pic in the last post of the bike set up for it pretty much shows why it is not a great idea - better to stick with barend shifters on mustache handlebars