Am I authorized to wear my home unit's SSI-FWTS patch if I'm with an Air Force unit in Qatar and spend any time in an IDP designated zone?

This is one of those questions that always seem to have multiple versions of an answer. Even though you will be serving under the Air Force, you will need to seek USARCENT guidance regarding Imminent Danger Pay and Shoulder Sleeve Insignia-Former Wartime Service. USCENTCOM has a number of named operations and exercises in their AOR, and USARCENT issues guidance to the Army on which of those (if any) would authorize wear of an SSI.
Based on previous ARCENT guidance, guidance which may be different now, SSI was authorized for Soldiers who:
1. Received both IDP and the Combat Zone Tax Entitlement pay
2. Were deployed on orders for combat or support missions to a USCENTCOM named operation authorizing a SSI.
* Check with USARCENT is my final answer.

COL Dan Ruder is correct; the listed criteria will need to be met in order to be authorized the wear of a SSI-FWTS (or whatever they are calling it now). However, it would not be your home unit SSI, unless they are deploying as well.
I had a very similar/same experience when I deployed to Afghanistan in 2020. I deployed as an 'individual augmentee' to a joint billet as the only member of the Army to an Air Force squadron. No one from my home unit deployed to the same operation. I was ultimately authorized to wear the USFOR-A/NATO Resolute Support SSI. I did ultimately receive a memo from my Air Force Squadron Commander, but was told by USFOR-A J1 that the memo was not technically necessary. When deploying with a different service branch, it will be your next higher-level Army-aligned headquarters that will determine what (if any) patch you are authorized.