Regulators can not maintain their set pressure exactly. The decrease in downstream pressure which occurs as flow rate increases is due to how the valve functions. All regulators function basically the same way. Some are spring loaded and some are gas loaded (called "dome loaded" regulators). Regardless of which type you have, the valves function the same, but for the sake of explanation, I'll explain it using the more common spring loaded regulator.
The spring acts against a diaphragm, pushing down on the diaphragm. This assembly also pushes down on the poppet that controls flow. The further down the spring pushes, the farther the diaphragm moves and the further open the poppet goes. The further the poppet opens, the more gas that can get through and the higher the flow rate through the valve. The gas pressure downstream of the valve is being 'measured' by the diaphragm with that pressure on the inner side and atmospheric pressure on the outer side. So the force on the assembly is balanced by the spring pushing down and the pressure pushing up on the diaphragm.
When you set the regulator at a certain pressure under a no flow condition, the pressure downstream is acting on this diaphragm and the spring is balancing that pressure by applying an equal force downward while the poppet is on the seat, stopping flow. As downstream pressure drops, force up on the diaphragm also drops and the spring extends and the valve opens a little bit. But a spring has a spring constant (k) so the valve will come to equilibrium with a slightly lower pressure downstream, balanced by the slightly lower spring force and with the valve slightly open. As pressure downstream decays even further, the valve has to open more and the spring has to extend more and the forces balance at this lower pressure and lower spring force.
So the spring constant (k) is what is affecting the balance on the diaphragm. If you have an infinitely small spring constant, there would be no decay in downstream pressure with an increase in flow rate. But there is no such thing as an infinite spring constant and real regulators experience a decay in the downstream pressure set point as flow rate increases. This phenomenon is sometimes called "droop" although I've heard manufacturers call it by different names. It's a function of the spring constant and diaphragm area.
If you need a regulator with minimal droop, try a dome loaded regulator. The effective spring constant of a large volume of gas is generally lower than a spring.