There are ways to do this transparently if you control the machine doing NAT for the classic computer. Doing things that way used to be more common before https became the default for most sites, because nowadays you'd have to install TLS certificates on each client to not constantly get errors. However, if you're also proxying https to http, then no worries!

I should update my transparent proxy how-to to be used with MacProxy Plus...

The browser I'm using (MacWeb 2.0) has its own proxy config (you can see it in the demonstration video around 3:27), but you can also enable a proxy at the system level using Internet Config (click Firewalls, check "Use HTTP Proxy", then enter the server address and port in the text field).

It's been a while, but from what I recall with early Macs apps like mail clients and browsers would have their own proxy settings rather than any system-wide setting, so you'd set the proxy in Netscape (or whatever).