MCU HIB pin only pulls Buck/Boost EN pin every-other Hibernation
I suspect that in step (4) the Vbat supply is drooping too low and the MCU is entering brown-out, hence the dim LEDs (GPIO in reset tends to drive LEDs weakly). I suspect it's doing this because according to the Tiva datasheet, the GPIO remains active during hibernation, so that LED you have connected to GPIO doesn't necessarily shut off as you claim.

So as you enter hibernation, the HIB pin will go low, causing the 3.3V to fall. The MCU will switch to Vbat but because there is a 51R in series, it will droop trying to supply the GPIO which is still on. It'll hit brown-out, and some oscillation could occur.

Maybe the second time around the GPIO has switched off and so the switch over to Vbat succeeds.