This is a tough situation. I've read from many reputable sources (e.g. Chronicle of Higher Ed.) that some schools may permit their finalist to delay start for a year to complete additional training. But, this may vary tremendously depending on the institution. For someone who is a "hot" candidate with impressive research and receives an offer from an R1 institution, I can see this as a possibility. However, for any more teaching-oriented college/university, I'd be skeptical.
At most teaching-focused institutions, or those that have a balance between teaching and research (e.g. 3/2 or 3/3 teaching load), faculty slots are often replacements for someone who's leaving. Often these institutions need someone to pick-up the departing faculty member's courses right away.
What might be a safer bet would be to first request a delay in when you need to make a final decision on the postdoc. Then, wait to see if offers come through from the TT positions you've had interviews for. If needed, you could try to "nudge" them by honestly noting that you've had a solid post doc offer that interests you. Now, my suggestions here are assuming that you'd take any of the TT positions over the post doc. If the post doc is so attractive that it outweighs the TT positions, then go for it.