You'd have to do a bit of Assburgering to get the Mikuni to fit the Briggs- they have different mounting types. I found a whole bunch of adaptable mounting flanges in the DennisKirk snowmobile catalog they sent me with my order.
As far as carb size, here is my Assburger Estimate: 20mm. Based on my wife's carbs on her bike. Bear with me....
Each 20mm carb can handle an 87cc cylinder all the way up to 9000 rpm, no problem. Your motor is about 138cc. I took the cc's that the 87cc cylinder is flowing at 9000 rpm (783000cc) and divided it by your engines cc capacity, and came up with a potential 5650 rpm.... which means even the 20mm carb is a bit on the big size. (On a 5hp B/S, it'd be good up to 3600 rpm.) They sell the 20mm carbs on Ebay for $30. You'll also have to mount the gas tank higher so that it'll gravity feed. I'd think this carb and about a 12" straight pipe on the exhaust would do pretty well.