First off, I am not a medical professional, just a CPAP user that takes some interest in the technology of the machines. My observation is that the apnea treatment field is in transition. The old traditional approach was to do an in lab overnight PSG. If it was positive for apnea, then the next step is an overnight titration step in the lab to determine the fixed pressure for a CPAP. You are sent away next with a trial machine set up to that pressure, and after a week or a month then you bring them the results for any tweaks required to the pressure. Now with the auto CPAP or APAP so common and inexpensive, I would say most clinics would simply set up the machine with a wide range after the positive test and jump to the at home trial stage. And the even more later development is the use of the ResMed and Phillips at home sleep study equipment. Labs here in Canada that use the at home approach, would have gone from your first positive test of moderate apnea to the at home trial of an automatic machine.
This process gets complicated a bit by the various jurisdictions, the US Medicaid system, and insurance companies. They can force a stepped approach; Sleep Study in Lab - Titration Test in Lab - Fixed CPAP - At home trial - If that fails, then APAP, at home trial, ..... If you have insurance and are essentially paying a substantial amount it makes sense to me to cut to the chase for economics and expediency. The condition however is that is in the absence of complicating factors such as heart and lung disease or other issues. The basic no complications process leaves out the in lab tests and titration and just jumps straight to an APAP.
If you want to go that route and you can see a GP soon, perhaps just ask the GP what he/she needs to write a prescription. If the system is the same as Canada, the GP can request a copy of the report from the sleep specialist and use that to write a prescription for a machine. I believe the same as in Canada courts have determined that medical records belong to the patient and the medical profession has no right to withhold them from you.
As for a machine the suggested ResMed AirSense 10 AutoSet is a good one, probably the best on the market. It comes in a For Her version which is the same price, but includes an extra optional mode that has been optimized for women and potentially can work well with men too that have lower pressure needs (<12 cm). I have the standard AutoSet, but if I was buying again I would buy the For Her version (despite the color and flower pattern!).
Setting the machine up is not that hard, and you can monitor your results as you have already determined with SleepyHead. Assuming you are in the US it pays to shop around for prices on machines. There is a place called SecondwindCPAP that sells both used and new machines at good prices. There may be better places. I am from Canada so have only shopped seriously here.
The final issue to consider is that not everyone adapts well to using a CPAP, and typically the mask is the biggest issue. Here, the sleep study companies offer a free trial of a machine for two weeks. They also put ridiculous prices on their machines (like $2400 for a machine and mask). But, here at least you can do the trial as I did, then return the machine and buy the machine and mask on line for $850 or so. I am assuming if you go down the GP prescription and buy it yourself route, then you have bought in, and can't back out. There are some companies such as CPAP.com that offer return insurance on some things like masks, but I'm not sure about machines.
You may also want to contact your insurance company and make sure they are on side with your course of treatment. Generally the lower the cost to them, the happier they are, but when you deal with bureaucracy you never know!
Hope that helps. If you have any questions, just ask...
Here is a link to an article that discusses how the apnea field has evolved, and the reasons around it.
AutoPAP vs CPAP