Hate to say it, but you are going to HAVE to drop qualifiers or increase your budget (or go used).
The only new as-built machines that will meet that kind of budget are the aliexpress machines.
You COULD build your own machine, but you’ll have to get creative with parts, and you may end up having to go 2U.
Rackmount hardware is always going to have a premium attached to it it seems.
Spanning tree protocol will kill any concept of “ring topology” and quickly for good reason. You’d end up with broadcast storms overloading and killing your network very quickly.
Pick up a networking fundamentals book. As usual, Oreilly is good for this (https://www.amazon.com/TCP-Network-Administration-OReilly-Networking/dp/0596002971?ref=d6k_applink_bb_dls&dplnkId=3456f4a6-4e65-40ac-a6fb-9a05e4796c1b#customerReviews)
Anything on the same vlan and subnet will be able to access eachother without additional routing. Only upon crossing vlans and subnets will you need routing.
If you are wanting to increase capacity between switches, use multi-gig backhaul links, and leverage LACP if needed. Male sure that the switches you are using are capable of switching the throughput you need.
I’d be very surprised if you are saturating 10g/40g links, or hitting switch max throughput numbers though.