Using AP-A with multiple "special" devices

So, you're wanting to know how to configure digital phone service to an RJ31X jack so it controls the rest of the network should the need arise. Correct? I'm not going into further detail until I know for sure that's what you're asking. Also, you need to start with checking with the alarm company to see if they support digital phone service, most don't. 

Assuming you are no longer using the PSTN connection, you would need to run the POTS output of the AP-A to the wiring from network interface as it enters your house or at some point between the network interface and the RJ31X.

You would also need to make sure the PSTN interface from the telco has been removed from your current house wiring. This should be anyway done for non-alarm usage if you plug your digital phone service into any wall jack as the disabled POTS network may still have power (but no service) running thru it and could damage or interfere with the AP-A (which is one reason for all the warnings about NOT plugging it into an existing jack in the self-install instructions).

Even then this is fraught with potenital issues (e.g., does the alarm service work with your digital phone service).

It may be easier to migrate your alarm system to cellular.

Just to get the acronym right it's ATA. 

AP-A is a valid acronym and different from ATA.

The first refers to a specific device and / or the associated service from AT&T known as AT&T Phone - Advanced.

ATA referes to analog terminal adapter, the generic term associated with a digital-to-analog phone adapter (which can be either a standalone device dedicated to that function or a software component embedded within a multiple-purpose device, part of whose function is to connect an analog phone to the digital service).

"AT&T Phone - Advanced" sounds like a service offering, and I think it's really supposed to be the name of a service offering.  But, apparently it comes with just one implementing device, as shown in this setup guide:

https://www.att.com/scmsassets/support/other/att-phone-advanced-online-guide.pdf

So, it's a VOIP device that can hook to your Internet, and use the AT&T Wireless network as a backup (or just use the AT&T Wireless network to provide analog phone service).  This is similar to other VOIP solutions that allow wireless backup (c.f. Ooma Telo LTE).  

In any case, the RJ31X needs to provide the correct Tip and ring on the inner pair of wires connected to the jack on the Telco device, whether that be regular old ATA, a jack on the AT&T Gateway, this cell-modem/router/ATA combination, or what.

That's all news to me. Analog phone jack with cellular back up, but still the same service as far as what the op was looking for as @JefferMC points out.  ATA with a wireless home phone back-up or use it as a wireless home phone only. Thank to both of you for pointing this out. It could still use the homes wiring just like any ATA or wireless home phone when wired correctly. I'd agree it's more of a service than an acronym.

It could still use the homes wiring

Per the guide "NEVER plug the AP-A phone cable into your home phone wall jack."  However, of course you can, so long as you observe all the requirements for backfeeding dial tone.

FWIW, a long time ago when switching from POTS to what was then called Uverse, Att configured my internal home phone wiring to work with the alarm system using an ATA.  Presently I have MagicJack service connected to the circuit.  

In addressing the OP's question I'd think using an ATA (analog telephone adapter) along with cellular back-up would be an added issue with an alarm system. I'd go as suggested by @thechef1 with "It may be easier to migrate your alarm system to cellular"