LDO Voltage Regulator - What happens inside dropout range?
It's an emitter follower with voltage and current feedback, so it just follows the input by the same dropout voltage if at the same current.
Due to saturation of Vce at dropout threshold, Rce resistance rises a bit and current gain has reached a minimum ~10 so voltage dropout tends to increase with more current.
But a 1V dropout is unsuitable for your application, so search for a &lt;50mV dropout LDO using FET technology rather than BJT's. There are many.