Assuming your 10K isn't for purchasing a capital asset and is just for actual flying costs/spending money I think I'd get trained up on a Cirrus (or other competent machine) and then simply set aside a few weekends per year to take the Mrs/girlfriend/boyfriend/mistress somewhere in Europe for a long weekend in a nice hotel somewhere.
Flying is expensive, so if you're not going to do much of it why not make sure that every flight is a memorable one?
Sure your currency isn't going to be the best but within your budget you could have an hour with an instructor once per month to stay on top of things.
If you won a competition and the prize was a weekend for two in a nice hotel in some romantic European city, transport by private aeroplane included, you'd be delighted wouldn't you?
So just set the competition yourself and make sure you win!!!
Life is too short to wait for someone else to give you the prizes.