Is one to invoke blessings upon the Prophet (peace be upon him) if he comes across a verse that reminds him of it during prayer, or is invoking blessings upon him restricted to the Tashahhud only?

He should not do that in the obligatory prayer, because it was not transmitted from the Prophet, peace and blessings be upon him. As for the supererogatory prayer, there is no harm in it, for he, peace and blessings be upon him, used to pause during his night vigil at every verse that contained glorification and would glorify, and at every verse that contained seeking refuge and would seek refuge, and at every verse that contained a request and would ask. Sending blessings upon him, peace and blessings be upon him, falls under this category.