Do you get 4.8V while powering up the board? I am wondering if they cannot provide the inrush current needed by the board, causing a turn-on voltage dip which results in the power management IC to shut off everything. I cannot find a proper datasheet for the batteries, but I would expect them to work OK for this based on this little information. Could you try using the BAT.VIN pin (P2.14) instead? And/or try with P1.05 but with the LiPo stack and 5V SparkFun regulator you showed in the diagram above