Accepted post-doc and have subsequently received offers for full time faculty position - quandary
Maybe I shouldn't start the answer with the judgey part, but if you don't feel like you can accept the faculty positions because of your previous commitment, why the hell didn't you pull your name from consideration the moment you accepted the European job?  On my personal list of academic job hunting sins, not withdrawing your name from a position you've decided you can't accept is much worse than declining a postdoc offer for a TT.  So, there's really no ethical choice at this point, so you may as well do what's best for you long term.

That said, I wouldn't be so quick to dismiss the possibility of starting the TT job with a delay.  This is very common in mathematics (my field); I've done it, and I know dozens of other people who have as well.  I can imagine reasons it would be harder in mechanical engineering, but it's worth a shot.  An important thing to remember as a job candidate is that once you have the offer, the tables are reversed.  They've shown their hand, they've made a time, financial and psychic investment in you, and their other candidates are slowly slipping away as they wait for your answer.  They really want you to say yes, so a concession like letting you take an initial leave is a small one.  After all, they don't even need to pay for it!  Universities are very flexible about letting you do things if they don't have to pay for them.

I mean, really they should want you to do this, since you'll come to them better trained, with more fresh ideas, and a wider perspective.  So sell it on those grounds, and I bet you'll succeed.