Can someone tell me what the cutout on the left is for in my bachi case?
I know the rectangular cutouts are for koma but the one on the left seems oddly specific in shape, but for what? Yubikake? (This is a double case.)
Also, should bachi be unable to move when strapped in? Bachi vary in size so I think some movement is to be expected, however the edge protectors don’t fit at all so I’m more aware of how much the bachi can slide around. Is there anything to look out for or be cautious about?
Hi-ho
Here is a diagram
Although slightly different, this image suggests its for yubikake.
Your bachi may shift around a little if it isn’t a perfect fit or if it is rather heavy. If you are concerned about damage, please continue to use a saya on the blade.
Thank you for the labeled diagram. 
Neither of my saya fit inside the case. 
Is it normal for saya to be wider than the bachi by 25mm or more?
I could trim down the wooden one but before I break out the wood saw and sandpaper are there any alternatives or DIY solutions?
While Bachi has a very wide range of blade sizes, from Tsugaru at 95mm (yet I have seen ones that are 75mm) to Jiuta at 130mm (though I have one that’s 137mm), the saya is likely manufactured in a few standard sizes.
However, there is the possibility that the store you bought the bachi would stock one size or two for volume discounts with the manufacturer, instead of stocking a wider range of sizes.
Although 25mm is a very big difference, I believe that 10mm (5mm per side) is a healthy margin to have. But if it’s very snug, maybe 2 or 3mm will do.
So, yes, you can trim it, but leave at least 3mm per side, 5mm per side being ideal.
And if it’s not snug anymore, I have a nice trick for that. Let me know if you need it.
Regarding the bachi being loose inside the case, my ideal solution is to make a snug-block for the handle end, that will rest against the case, and push the blade snugly into the inbuilt saya on the case.
And even if you trim your saya to fit inside the case, your bachi will possibly still be loose, so you can make a snug-block anyway.
The block can be anything from a cube (like a 6-sided dice) made of cardboard squares glued together, to anything fancier, like a wood cube covered in thin foam or EVA, and then covered again in some nice Japanese fabric. So if you break out the saw, you can make a pair of snug-block blanks.
Inbuilt saya? Are the blade edges meant to push into the ends?
Great idea. I will make something a little nicer later on but for now a non-latex makeup sponge does the trick. 
Most bachi hard cases will have an inbuilt saya. Check this picture.
Wow, nice idea!!! I will try the makeup sponge! They come in handy cubes.
None of my cases do!
All of mine do, except the huge ones for Jiuta bachi.
I guess I just got lucky, because all I have was bought second hand.
Wait, have you bought Tsugaru-specific bachi case?
Stage-class Tsugaru bachi has that slope in the blade, and when it lies flat, the blade doesn’t touch the surface (see photo below).
So it wouldn’t make sense for a Tsugaru-specific case to have an inbuilt saya.
I am not sure why you’d call that stage class - it’s a feature of tsugaru bachi 
My bachi cases run in various sizes. Some are older, some are newer. None have that.
To differentiate from training/practice bachi, which are standard geometry (no slope) wood bachi, at the correct dimensions and weight.
I… don’t think that’s a thing (for tsugaru) - but that’s not relevant to this thread.
I bought my case secondhand and the original owner didn’t specify anything beyond “double bachi case.” I have no idea where (eta: I mean what shop) it came from or how old it is, though given the condition I think it’s pretty new. Currently the make-up sponges are doing a great job of keeping the bachi from sliding around so I’m not breaking out the tools yet.
I have one very similar to what you have, bought a few years back.
If its the same maker, take extra care with the pins holding the hinges - they fall out unhappily fast.
The hinges are surprisingly small and delicate given the retail cost of these cases. 
I will keep an eye on it though I wouldn’t mind replacing the hinges with something slightly more robust. I’d also like to change out the material since I have a nice floral yukata going unused.
To keep the pins in, buy superglue that’s suitable for metal.
Those are usually more fluid than viscous, but if there are viscosity variations, choose one that’s more fluid but not super fluid, as you want some penetration but not leaking all around the hinge.
Apply a tiny drop (if possible with a micro-tip attached to the tube) to one end section of the hinge, allow it to penetrate that section, and let it cure placing the case in a vertical position, with the hinge section with the glue at the bottom.
It surprises me that the manufacturers don’t lock in the pin on the first section, to prevent this kind of issue.
