Haha, another topic asking if the weight limits are strict... Well they are certainly there for a reason! Perhaps you could get away with 10% more or let's say 9kg, maybe even 10kg on a "good day" (though the trend is, that they are getting stricter and stricter), but nowhere near 13kg.
Questions/answers :
1. Never heard or read such an argument for buying up to business class before. So no! Just book a checked bag for much less!
2. Yes, precisely. You can check out the bidding options (the minimum bid) without actually having to place a bid, so go ahead and do that. I still wouldn't pay more than what a checked bag would cost for the upgrade, given fast track and lounge access are already available (don't know about the latter in your case, just saying).
3. Document everything you have in your bag and even take a picture before you close it. If it's lost or damaged you are entitled to compensation and I think that's (up to) a four figure amount. Unless you are only going to spend hours on the island (which does not seem to be the case) just book the checked bag and save your $$ for a nice meal. I've had my "priority" tagged, business class checked bags arrive LAST on the belt 3 times a couple of years ago (always at Athens Airport). That took about 45-50 minutes, so fyi that should be your worst delay-case scenario, even if you travel in economy.