Hi. Don't claim to be an expert on this kind of thing, but as no one else has replied yet...
The S exhaust will fit as quite a few people have done it, but you'd need to change the rear valence as well so that you have a gap int he middle rather than the side! i guess there will be other modifications required too just to hang the thing.
£300 sounds very expensive just for the back box. have a look round some of the sponsers of the site. just had a quick look and Minispeed do a complete Milltek exhaust for the cooper for £200 + vat. Don't know how much fitting would be though! just thinking you could use the opportunity for a mod!
New Mini Product | Milltek side exit exhaust system for Mini One & Cooper - 90mm special tailpipe
(just looked at the sponsor list and minispeed weren't on there? Are they not a sponsor? Thought they were. If not - apologies Paul!)