The Samyang AF 12mm f/2.0 E is definitely your safest bet honestly.

• Lightweight for long walks
• Decent AF for talking heads
• Fits the budget perfectly

Honestly, Ive been filming travel content for years and people always get too hung up on f1.4 lenses. For a travel vlog in Japan, you actually want some of the background in focus so people can see the scenery. If the background is just a blurry mess, you might as well be filming in your bedroom. Directly to your point about the Viltrox AF 13mm f/1.4 STM Sony E-Mount... it is definitely chunky. On a tiny camera like the ZV-E10 II, it feels very unbalanced and front-heavy. I wouldnt want to carry that around Shibuya all day, especially handheld. It is a great lens, but maybe not for this specific trip. Heres what I would actually suggest for your $350 budget:

• Buy a used Sony E 10-18mm f/4 OSS. You can usually find these for around $300 to $330 on the used market. It has Optical Steady Shot (OSS) which is huge since your camera doesnt have internal stabilization. Itll make your walking shots way smoother than any prime lens will.
• Look at the Meike AF 12mm f/2.0 Sony E-Mount. Its surprisingly light, fits your budget brand new, and the autofocus is snappy enough for talking heads. Dont stick with the kit lens. 16mm is just too narrow for handheld work and youll end up with a video of just your face and no Japan. Just make sure you order soon so you can test the autofocus before your flight... nothing worse than getting to Tokyo and realizing your gear isnt working right.

Just saw this and man, I feel your pain. I went through three different lenses before my last trip to Kyoto and honestly, most of the popular gold standard stuff is a total disappointment for actual travel vlogging. I had the Sigma 16mm f/1.4 DC DN Contemporary and it was just... way too much. Not as good as the YouTube hype once you are actually walking 20k steps a day through Shinjuku. I tried it on my ZV-E10 and the lens is literally twice the size of the camera body. It felt like I was holding a brick. Unfortunately, the Viltrox AF 13mm f/1.4 STM also had issues for me. While the image is pretty, the autofocus had this weird micro-jitter that ruined my B-roll. Nothing worse than getting back to the hotel and seeing your footage pulse every five seconds. Quick question tho... what kind of setup are you using for the actual handle? Just the camera body or do you have something like the Sony GP-VPT2BT Shooting Grip? Because that weight balance changes everything. Since your budget is tight, you might have to look at these:

• Sony E 11mm f/1.8
• Try to find this one used. Its the actual king for vlogging because its tiny, but brand new its over your $350 limit.
• TTArtisan AF 14mm f/3.5
• Its cheap and small, but the aperture isnt great for night shots in Japan. Really bummed to say that the Sigma 16mm is probably gonna be a mistake for your wrist. Its a studio lens masquerading as a vlogging lens.

Quick reply while i have a sec because omg i am literally in the exact same boat right now and it is driving me insane!! i have been shooting on sony for years and i love the system but i have been scouring every corner of the internet for weeks trying to find the perfect wide lens for my Sony ZV-E10 II before my trip and i am just as stumped as you are. honestly i think this camera is fantastic but the lens struggle is so real when you want something light that wont give you carpal tunnel while walking through akihabara. i have been dealing with this for a solid month now and every time i think i finally found the one i see a review saying the autofocus is trash or it weighs a ton... it is seriously the most frustrating thing ever i just want something reliable for my travel vlog and i am gonna lose it if i dont find something soon!