Just ran some tests again.
You need 276/277 Piety, which amounts to 269 base Piety + party PIE buff.
The specific condition where the rotation is possible is when you have full mana, use F3 to switch from UB3 to AF3 and you get another mana tick after UB3 wears off.
During the first Enochian rotation (30s), it is doable even when waiting for a full mana tick.
During the second Enochian rotation (25s), you will have to Swiftcast B4, else you will lose Enochian. If you're able to use this rotation during both rotations, there is no need to Swiftcast B4, since Enochian CD is only about 5-7s.

Another sweet side effect is: If you have full mana, use B4 and then go into AF3 with F3 without an UB3 mana tick (which means you have [(Full Mana) - (B4 cost) - (F3 cost)] ), 277 Piety will leave you enough mana to use B4 after casting F4 > F4 > F1 > F4 > F4 > B3, which is normally not possible, thus saving a mana tick.