There are scores of SMTP providers you can find online, and almost all of them have a free-tier which doesn’t require a credit card.
To expose the service to the internet, you can use something like Cloudflare tunnels (Free), without opening up a port on your router.
But this is a huge security risk, as it allows an adversary to exploit vulnerabilities on discourse or anything else you expose using tunnels to establish a foothold on the server, and completely compromise your home network. If you end up doing this, at the very least you might want to place your server in the DMZ network of your router, and not open any ports to it. Do any admin work directly on the server.