I'm looking for PC horizontal case that will fit into this Furniture hole:
It is hard to find one. So I'm looking for the horizontal PC case that will support:
140mm is 5.5", this only leaves 3/4" of space to the top not counting case width. I don't think that is do-able, but you should be able to squeeze in a 120mm AIO setup.
Perhaps the following case: https://www.newegg.com/Product/Product.aspx?Item=N82E16811144230
It stands 5.3" (134.6mm) high. This gives you some vent room on top, and, it will accept 2 120mm fans.
When you build in a restricted space like that, you wanna let air come in and go out from every direction possible, otherwise you get heat pockets. Doesn't matter if you run a rad or not.
Also, if interested, I did a small build guide for doing an ITX in a case that might be suitable for the area in question: https://pcpartpicker.com/guide/h6ZxFT/nmediapc-1800b-htpc-mini-itx-build-guide
As I am sure you are away this is going to be an incredibly niche case to find. I think the best way to do this would be to ditch the AIO and use something like low-profile like a Cryorig C7. It is also going to be easier to use ITX as there are a ton of cases that would fit this need. Something like the Node 202 would be good.