Thanks for your feedback, looks like this is a harder problem than I was hoping... I have already purchased the chassis so am a bit stuck with it unfortunately. By getting the deeper 4728 version, I hoped there would be more options to work with.
Since I was unfamiliar with it, I googled it and the 4728 appears to be a lightly longer version of the 4712. It didn't look promising in my opinion, but actually trying to make it fit with the actual case in front of you will give you a much better perspective. Maybe you can work something out.
In that case, is it possible that I can mount these cards somewhere internally (eg. even with some custom mount/3D printed thing so they don't electrically short out) since they don't need external cable access and their thermal requirements should not be too extreme?
Could be possible. Again, since you're going to have that case in front of you, you have the best overview of the situation. From experience and by looking at it, I'd say no. But that shouldn't discourage you, there's no substitute for sticking your head in and look around in the live thing.
Another idea is to route at least one of the 4090s with a riser cable and find somewhere custom to mount that in the case, thus only blocking one slot at least. I have heard vertical mounting of GPUs is possible, for example, but am not sure if this makes sense in my chassis?
Vertical mounting means the GPU is parallel to the board), instead of the usual horizontal position (90° angled slotted into the board). This is usually done for style, not for functionality. Some tower cases allow for a half style, half functional vertical mounting. For example, the Fractal Design Meshify 2 XL allows for vertical mounting of a 3-slot GPU while still allowing full height PCIe cards to be put in behind it. Another case, I think from Lian Li, allows vertically mounting the GPU in a completely different position, not overlapping with the board at all. But for most cases, it actually means mounting the GPU as the only PCIe card, with a riser, slap bang in the middle of the PCIe slot area, effectively blocking ALL slots. But this is all academic anyway, as it doesn't apply in your case (pun intended).
In some 19" cases, functional vertical mounting of GPUs is actually the default, because GPUs don't fit upright in 2U cases, for example. You can usually get about 4 2-slot GPUs in an industry standard 19" 2U server from Dell and the other usual suspects. Supermicro might have dedicated GPU cases for standard form factor components that do the same thing. But again, nothing of that is of value to you since you're already getting that case.
Two additional options I can think of:
1. 1-slot water blocks for the GPU, but that means you need to fit a pump, a reservior and radiators. Some of it might have to be external.
2. Talking about external, external GPU cases.
I guess none of that is really what you imagined with this build.