I would suggest looking at the Sony ECM-B10 Compact Shotgun Microphone for your A7SIII. I dealt with a similar interference nightmare during a crowded wedding in a hotel ballroom where the wifi signals just trashed my analog cable signal. Since this uses the digital MI shoe, that whole issue goes away. You might want to be careful with the auto-gain feature tho, as it can sometimes pump in quiet rooms. I usually lock it to manual and keep the attenuator on 10dB if its a loud reception. Make sure to check that the pins on the shoe are clean before your NYC shoot because if they get dirty, the camera wont even see the mic. It fits right in your budget and is way more reliable than the older analog Rodes.

Just saw this and unfortunately, my experience with the budget MI shoe mics has been pretty disappointing for pro work.

• Sony ECM-G1 Ultra-Compact Shotgun Microphone: Lacks manual gain dials, making it hard to stop clipping during loud wedding receptions.
• Sony ECM-XYST1M Stereo Microphone: Build quality feels cheap and it picks up too much handling noise in my experience. Both felt like a step down from my old gear tbh.

I've been really happy using the Sony ECM-M1 Shotgun Microphone on my A7SIII for the last few months. It sits right around 340 bucks but honestly you can probably find a deal or just stretch the budget a little for the quality you get. Since you're dealing with NYC interference, the digital connection on the MI shoe is basically mandatory... it completely skips the internal preamps and uses the camera's digital interface so you dont get those weird buzzing sounds from phone signals. I have zero complaints with how it handles loud environments. It has a physical gain dial on the back that is super easy to thumb while you are filming, so you can stop clipping before it even happens. It kinda saved my audio at a super loud reception last month where the speakers were way too close to my rig.

• 8 different pickup patterns including a rear-facing one for when you're narrating
• No cables or batteries to mess with since the camera powers it
• Low-cut filter helps a ton with that low-end rumble in big halls
• Physical switches on the back for quick adjustments during the ceremony Just make sure you have the switch set to Digital rather than Analog on the side of the mic body. It works well and it is tiny enough that it wont throw off your balance if you are using a gimbal for the reception. Good luck this Saturday, NYC weddings are always a trip.

Ngl, moving away from analog cables saved my sanity at loud receptions last year. I switched to the digital connection for my A7SIII and havent looked back. Just curious tho, are you mostly doing run-and-gun b-roll or do you need something that can handle a backup safety track for the vows? Also, how close are you usually getting to the speakers during the dance floor madness?