So, it looks like my poor Mini has developed a problem with the idler gear bearing or something closely related. When researching, I came across this little gem from Keith Calver on the Mini Spares website. Keep in mind that the classic Mini was in production until 2000. So that's 8 years of using whatever shims happened to be in the bin.
Mini Spares said:End float is a continual problem as folk either ignore it through ignorance or lack of accessible information on how to do it, or belief special tools are needed. Also, later factory assembled engine units (from about 1992 onwards) were built up using whatever shims and thrust washers were available, since Rover were not making regular orders for all shim/washer sizes due to the forthcoming end of production. Consequently, many units left Rover with incorrect (usually too big) clearances. The idler gear was no exception.
http://www.minispares.com/Articles.aspx?ty=ad&aid=107
On a lighter note, I parked it beside the Dodge last night. Yes, the front bumpers are in line.

