How can I pursue an exception to policy to avoid separation from the USAR for not yet having completed AIT before 24 months TIS?

PFC (Join to see), after reading through the comments and amplifying information you provided, I figured I’d give you ground truth on some of the aspects of your situation. Keep in mind that the basis behind your separation for not completing IET within 24 months is something stipulated by law (10 USC § 10105, Section 1115(c)(1))*.

First and foremost - you aren't going to get an exception (or rather, the likelihood is extremely low). If you can show that there was a gross injustice done to you, or some other mitigating circumstance that impacted on your not being able to complete your training within the 24 month requirement (e.g., showing that you were the victim of misconduct and someone was deliberately preventing you from being trained; etc.), it might be an outside possibility, but not in the situation as you described.

If you're still in your entry-level status, then you'll receive an Entry-Level Separation (ELS) unless you are being separated for something other than minor misconduct or unsatisfactory performance (AR 135-178*, 8-1a). For a reserve component Soldier, your entry-level status doesn’t end until 365 days after starting ADT for your IET. If your IET is split into two or more ADT periods, then your entry-level status will end 180 days after starting your second ADT period. As you stated you still haven't started AIT, you're still in an entry-level status.

There is no characterization to an ELS (AR 135-178*, 8-3). Although it is possible to be separated during your entry-level status and receive a characterization for your service (normally negative), you wouldn't receive an ELS and would be separated under the appropriate chapter (i.e., serious misconduct, criminal conviction, etc.). In other terms you can think of your service as being unofficially characterized as “other than dishonorable”.

Your time in the USAR will still count and if you reenlist it will be used as creditable time when things such as your PEBD are recalculated. Your service for an ELS may be uncharacterized, but an ELS doesn’t mean your service doesn’t count. In the world of recruiting, you are categorized as a “glossary non-prior service” individual (prior military service but doesn't meet the classification of "Prior Service").

Having an ELS doesn’t prevent you from reenlisting, but an ELS discharge will be given a reentry code of RE-3. If you have an ELS, you will be prohibited from reenlisting in the Army for six months (from the date of discharge) and you’ll require a waiver approved by the CG, USAREC (RA or USAR) or the Director, ARNG (AR 601-210*, 4-13b(3), which is likely why the recruiters have been non-committal.

You will still be the grade you are separated at, but keep in mind if you reenlist, you’ll go through a grade determination according to AR 601-210 and USAREC Reg 601-210. That means you might not be reenlisted at the grade you separated at, or if you are, the date of rank may be adjusted to the date of enlistment.
--------------------------------
* 10 USC 10105 (Army National Guard of the United States: composition) https://law.justia.com/codes/us/1998/title10/subtitlee/parti/chap1003/sec10105/ - note: Despite the name, it does apply to the USAR (the 1996 NDAA changed the language from “National Guard” to “Selected Reserve”)
* AR 135-178 (ARNG and USAR Enlisted Administrative Separations) - https://armypubs.army.mil/epubs/DR_pubs/DR_a/ARN37322-AR_135-178-000-WEB-1.pdf
* AR 601-210 (Active and Reserve Components Enlistment Program) - https://api.army.mil/e2/c/downloads/344222.pdf

Here is my recommendation:

If you want to stay in and NOT receive the ELS, then I strongly suggest you pick an MOS that you can attend AIT NOW and then down the road somewhere you submit to reclass to CA. This is the fastest and least hassle filled fix you can get.

You want in? Then stop being like a horse with blinders on. I have been there. I fully understand your problem as I had problems enlisting, but I took what was available to me at the time, did that MOS for 2yrs, then moved to where I wanted to be. If you really want to serve, then sometimes you got to bend with the wind or you will get snapped off. Where there is a will, there is a way. You need to keep your eye on the prize, don't get stuck on the "I wants".