I totally get the frustration. I took my a6400 to the Swiss Alps last year and it was a mess for similar reasons. The battery life is unfortunately the biggest weakness of that body. I used to carry six spares in my pocket just to make it through a cold afternoon, and fumbling with them in gloves is just miserable. I almost sold the whole kit after that trip because it felt like the camera was working against me rather than for me. To fix the grip and the fumbling, you really need a baseplate. I started using the SmallRig L-Bracket for Sony A6400 and it changed everything. It adds just enough height so my pinky actually has somewhere to rest, and the battery door is much easier to find. For the strap, ditch that Sony branded wire immediately. I switched to the Peak Design Slide Lite Camera Strap and it distributes the weight across your shoulder instead of sawing into your neck. As for the screen, unfortunately, the a6400 display is just dim and not as good as expected for the price. I eventually gave up and just used the viewfinder way more. For the power issue, make sure youre using official Sony NP-FW50 Rechargeable Battery Pack spares because the third-party ones are even worse in the cold. Its kinda sad you have to buy extra stuff just to make the camera functional, but these items make it way more tolerable.

Re: "I totally get the frustration. I took my..." - honestly I think everyone who buys into the a6000 series feels this way at first. It is a tiny beast but the ergonomics are basically non-existent for hiking. I would suggest looking into a dedicated grip before giving up on it. I compared two different setups for my Japan trip and here is what I found:

• SmallRig Cage for Sony A6400 2310: This thing is a lifesaver for big hands. It adds just enough height so your pinky doesnt hang off the bottom. Be careful tho, it adds some weight and might make the camera feel less "travel-sized."
• SmallRig L-Bracket for Sony A6400 2331: If you want something lighter, this is the way to go. It solves the grip issue but stays out of the way. Just make sure to tighten the screw properly or it can wiggle. For the screen glare, you might want to consider the SmallRig Sun Hood for Sony A6400 2823. It works well but can be a bit fiddly to put on when you are in a rush. Also, I would suggest getting a bigger eyecup like the Kiwi Photos Long Camera Eyecup for Sony A6400. It helps block out the light when you use the viewfinder instead of the screen, which saves battery too! Just keep at it, you will find a rhythm...

Regarding what #3 said about charging inside your bag, i really wouldnt recommend that. I have had issues with overheating and it is just not as good as expected when you are on the move. It can actually stress the camera port if the cable wiggles. For your Japan trip, you need things that wont fail you. The a6400 is a great sensor in a frustrating body. Here is what I would do with that budget:

• Stick to Sony NP-FW50 Lithium-Ion Battery. Third-party cells often report the wrong percentage and die suddenly in the cold. To stop the fumbling, get a Think Tank Photo DSLR Battery Holder 2. It keeps them organized so you arent digging in your pockets.
• Neck straps are basically torture devices on ten-mile hikes. The BlackRapid Curve Breathe Camera Strap is much more reliable since it distributes weight across your torso.
• For the grip, look at the Meike MK-A6300 Pro Battery Grip. It is one of the few ways to actually get a decent handle on the camera and it holds two batteries at once. It basically solves the battery swap issue for at least half a day.
• Check out the website Photography Life for their a6400 setup guide. They have some specific power-save settings that might help you squeeze more out of each charge. Most people leave pre-AF or high brightness on and it just eats the battery. Turning off the screen when you are using the viewfinder is also a lifesaver tho.

The a6400 is a powerhouse sensor shoved into a body with zero ergonomics. Sony designs these for size, not long days in the field. If you are doing Japan in a few weeks, dont overthink it. Focus on weight distribution and power.

• BlackRapid Sport Breathe Camera Strap puts the weight on your shoulder instead of your neck. It is a total game changer for 10 mile walks.
• Meike MK-A6300 Pro Battery Grip solves the grip and the battery life at once since it holds two cells. It makes the camera feel like a pro DSLR and gives your pinky a place to rest.
• SmallRig NP-FW50 USB-C Rechargeable Camera Battery is easier to deal with than standard batteries because you can charge them with any phone cable you already have in your bag. Honestly, the screen is always gonna be bad in direct sun. Third party hoods are kinda flimsy and usually fall off when you are hiking. Just stick to the viewfinder and get a Hoodman Hoodeye for Sony A6000 Series. It blocks all the side light so you can actually see your exposure and focus properly even in noon sun. This setup keeps you under budget and fixes the actual pain points.

@Reply #1 - good point! i disagree about carrying spares tho. i would suggest a SmallRig Sun Hood 2334 for glare and a Nitecore USN1 Sony USB Charger to charge inside your bag.