I have a 8 sas drive ITX NAS. The one and only case that I found suitable is the Jonsbo N3 as it has a sata/sas back panel. It fits all of that in a small size so it is great.
It fits your price (not if you have to buy an itx motherboard tho) but it’s hard to find, even harder on second hand market.
(For the hardware inside it I went to used consumer hardware, like asus itx MB, intel 10500t and ddr4 non ec, for the same reason as you).
I got mine from my local eBay as 100% of the server is built on used hardware auctions.
The case took ages to find, mostly why it took me one year to build this server but I wasn’t in a hurry.