If one only has enough time before זמן תפלה to daven ברכות קריאת שמע or שמונה עשרה, what should he daven?
One should not daven Shemoneh Esrei immediately, but rather proceed according to the regular order of davening. After completing Birkos Shema, one may continue to daven Shemoneh Esrei until chatzos, though this does not fulfill the principle (rabbinic) obligation of tefillah.
Sources: Although the obligation of Tefillah may be greater than the obligation of Birkos Shema, Chazal instituted a specific order of prayer, with Birkos Shema coming before Tefillah. Because of this order, and because the times of both expire at zeman tefillah, one should not skip Birkos Shema for the sake of Tefillah.
This is implied by the Mishnah Berurah, who elaborates on what may be skipped when somebody is short on time, but does not mention the possibility of skipping Birkos Shema. See also Radvaz, quoted by Mishnah Berurah 90:28, who writes that a person must fulfill the mitvzah he is currently obligated in, even if this would come on account of another, greater mitzvah, that one would not be able to fulfill. This logic would apply in our case: one should fulfill the mitzvah of Birkos Shema even on account of Tefillah.