The Godox V1 has been my go-to flash for my A7III. The round head produces beautiful, natural-looking light and the recycle time is impressively fast. Plus, it's part of Godox's wireless ecosystem so you can easily expand with more lights later.

For your Sony A7III, the Godox V860III-S is an excellent choice. It offers TTL metering, high-speed sync up to 1/8000s, and a quick-charge Li-ion battery providing 480 full-power flashes with a 1.5-second recycle time. Priced around $209, it fits well within your budget.

If you're on a tight budget but still want something reliable, check out the Godox TT350S. It's compact, lightweight, and works well with the a7 III. While it doesn't have the power of larger flashes, it's perfect for event photography or as a fill light. Plus, it's part of Godox's ecosystem, so you can easily expand with more powerful flashes later.

Tbh I am always really CAUTIOUS about putting non-Sony electronics on my a7 III. I am still kind of a beginner and I worry about the cheaper brands maybe damaging the hot shoe or having weird software glitches. If you want to avoid having to DIY a fix later or deal with complicated tech support, I think the official Sony ones are worth the extra money. Here are two options that might fit:

• Sony HVL-F46RM
• This is about $400 and fits your budget. It is made by Sony so it basically guarantees the TTL and HSS will work every single time without errors.
• Nissin i60A for Sony
• These are known for being very reliable and they have been making flashes for a long time. Ngl I just feel better having that official professional service and warranty support. If a cheap light breaks, you are basically stuck trying to fix it yourself since there are not many self-service options for those electronics. Does anyone know if the third-party stuff ever causes actual damage to the camera pins? I would rather be safe than sorry.