Do pressure regulators restrict the mass flow rate?

I do that kind of thing regularly. My advice would be as follows:

1. Work out what kind of pressure you need in the tank for the maximum flow rate
2. Use a pressure regulator, as you mention in your question, rated to the pressure you worked out in #1.
3. Make sure the supply pressure to the tank is sufficiently above the maximum pressure you worked out in #1 and within range of the pressure regulator chosen in #2.
4. To control the mass flow rate, rather than using an orifice, use a mass flow controller on the outlet of the tank. This will ensure that you get the desired mass flow rate, while maintaining the correct pressure inside the tank. They can be expensive, but are well worth the money. I have had good experience with Bronkhorst ones.

Disclaimer: I do not work nor do I have any affiliation whatsoever with Bronkhorst, it's just from experience of using their products.

Your assessment of the situation is correct - the solution is of course to specify a pressure regulator with a large enough orifice that it does not cause you a problem.

Pressure regulators are specified with a Maximum Flow Rate, in the same way they are specified with Maximum Operating Pressure. You simply need to pick one that is suitable for your application.