300 NM X-country
Yes - it can be part of your hour building.
No - it doesn't need to be with a particular FTO.
Proof - log book entries will be sufficient but perhaps keep the receipts for the landing fees if you wish.
None of the log book entries for the CPL 300 miler that I've seen have been certified by anyone (unlike the PPL X-country which, as you know, has to be signed by the airfields you visit) and I've yet to hear of any being contested or questioned.
Others may have different experiences.
Good luck.