Personally I don't carry a tire, just a tube and I think its fairly rare to do fatal damage to a tire thats in good shape (especially for us relatively small light people - I ride a 44cm but I still have 700's), but it can happen. My husband did it just last week. He blew out a sidwall after he ran over a twig or something - it was a fairly worn tire and he gouged it. He patched it up with a powerbar wrapper until he could get to a shop that was fortunately pretty close- a dollar bill can work pretty good too.
If you've got a local shop that specializes in triathalon gear they will likely stock 650's since a lot of tri-bikes, even bigger ones use them.
Just an FYI, it might be worth it to put the tire on once at home before you need it. Depending on the tire putting a brand new one on in the field could be a bit difficult. I just put a new tire on (at home) It was a folding one, a Vittoria in fact and they do fold up fairly small, but the fit was so tight that it took both my husband and myself (him pushing and me pulling) to get the thing over the rim. I'm hoping that if I ever get a flat with this particular tire that having it on for a while will have stretched the bead a bit and it will be easier to get back on.