Update...
I pulled fuse 13, drove it to the airport and left it for a 5 days. I was worried there would be some parasitic drain, but the car started up just fine so it looks like fuse 13 fully disables the alarm system, and not just the siren.

I plugged the fuse back in this am and locked the car to see what would happen. The LED light on the mirror started flashing, but no beep when I locked/unlocked the car. Also, the alarm hasn't gone off yet, which tells me that the system is broken or needs to be reset somehow. My new theory is that replacing the ignition switch was viewed as a power interruption by the alarm system which, I think, triggers the siren. Pulling the fuse perhaps put the alarm system into a standby or other dormant mode that now it needs to be reset from...maybe...perhaps?

I'll plug the foxwell in and see if I can get a reset, or test of the system. Beyond that I think the only thing I can do is pull the negative from the battery and wait a while, which will reset everything.

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Current
Original owner 2002 E53 X5 4.4i to 4.6i swap
2026 G80 M3 6 spd
2025 G06 X5 50e
Former
1972 Audi Fox
1986 Saab 900S
1996 BMW Z3
1998 BMW E36 M3 Sedan
2004 BMW E46 M3
2006 Audi A3 Quatro
1993 Mopar 318 Jeep Grand Cherokee
2015 V6 Jeep Grand Cherokee