A DME is just a supplier of durable medical equipment (DME). There is no legal requirement for you to use one DME over another one of your own choice, nor any legal requirement that you stay with the same DME once you have purchased something from them.
Think of them kind of like a pharmacy. You need a prescription to buy certain items, but you can take your prescription and buy those items from any pharmacy you prefer. Some might be "in network" on your insurance plan while others are not. All of them want your business, but you might get a better price "in network".
As for machines versus supplies. Everything you buy is a separate transaction. You can buy some stuff at one DME and other stuff from another DME however you choose. If one DME doesn't treat you right, then take your business to another when it comes time to buy the next thing you need.
If your daughter purchased her machine then it's hers and she can go anywhere she wants for supplies. If she rented the machine, then there may be a minimum rental period (say 3 months) before she can return the machine, but even then she can certainly rent a machine from one DME and buy supplies from another. She certainly won't have to return a machine in order to buy supplies from somewhere else.
She might want to return the machine and get the "for her" model she wanted from this or another DME. If another DME, it may be more complicated since the first DME has surely filed with your insurance and there would be some red tape involved in undoing the first claim so the insurance can pay the second claim at another vendor.
Your insurance may have "in network" agreements or even contractual agreements with certain DMEs. So your choices may be limited by your insurance company if you want the costs paid by your insurance company within your policy. You mention a local and national DME call center, so if it's a big company then maybe you could choose a different local supplier within the same company if that big company is your only in-network choice.
You should be able to insist that the DME give you the "for her" machine that your doctor prescribed. Also giving your poor daughter a mask that doesn't fit because they didn't have one in stock is like buying her shoes that don't fit because one store didn't have her size in stock.
An ill fitting mask will be uncomfortable and won't make a good air seal so it will leak. It's already hard enough for some people to get used to a mask without adding discomfort and leaks from a mask that doesn't fit. In fact it is common for people to try a few different masks in the beginning until they find the one that best suits them. DMEs know this and most will allow you to swap masks in the first month without penalty.
Best resource for sorting out your options might be your insurance company. I would get on the phone with insurance and find out what your options are. Also let them know what this DME has done to your daughter by providing a machine that is different from the one she asked for and the doctor prescribed, and especially the business about the improper size and type of mask because they didn't have the right one in stock, then a month of followup to get it resolved.
DME is a highly competitive business, and they have to compete for your insurance company's business. They are not like doctors who decide which insurance they are willing to take. DMEs bid aggressively against other DMEs to win business from insurance companies. So your insurance company may take this DMEs poor customer service to subscribers into account when the contract comes up for renewal.
Bottom line is the best thing you can do in dealing with DMEs is to be an informed consumer. The DME may feed you a bunch of baloney or only share info or options favorable to them. But if you know your policy and tell them, they will line up with what you want to get your business.
And remember that the DME is not your insurance company, they are just a supplier who wants yours and your insurance company's business. They are a little different from a candy store because some of what they sell requires prescriptions, but they are not terribly different from a pharmacy (well, except that I trust what a pharmacist tells me more than I trust what a DME tells me