The site is probably just checking whether your IP address is in the known list of Tor exit nodes.
A pretty much identical question exists here : https://tor.stackexchange.com/questions/1229/how-to-chain-proxy-after-tor
While I somewhat question your motivation (are you attempting to bypass this block due to you actually being a spammer?), I'm going to err on the side of this being a legitimate request (and besides a website owner will still block whatever proxy IP you come out of if you abuse the site on the new IP address).
Most of the Tor proxy links I could find are for users trying to use a proxy BEFORE Tor, you need to use Tor to connect to a proxy server on the internet AFTER going through Tor.
As you mentioned something web-based you could visit while using Tor something like www.hidemyass.com to then appear to come out of the HideMyAss range instead of Tor.
The reason changing the proxy settings in your browser didn't work is because Tor behaves like a proxy server, so removing the Tor settings and providing just a normal proxy stopped it from going through Tor (unless that proxy itself goes through Tor).
I don't know if this already exists (this site suggests it does : https://www.deepdotweb.com/security-tutorials/chain-socks-tor/) but making a program to behave like a proxy server that then connects to a proxy server via the Tor proxy server would work, you would specify this relay as the proxy in your web-browser and then leave it to this new program to set up the connection to the proxy on the other side after going through Tor.