Can I fake a static outgoing IP address?

A client has a control panel that is web accessible with a whitelist of IP addresses that can access the page for security reasons. I called my ISP to see about getting a static IP address thinking it would be a simple add-on. Unfortunately they only offer static IPs to their business customers, and the cheapest business package is significantly more than I pay now and provides a slower connection. It doesn’t seem the best solution.

Is there some networking magic or a service that can make my outgoing traffic appear to be a from a static IP? That way I can add that IP to the whitelist and not have to worry about updating it. Is there some way to “bounce” my traffic off of an external server so that IP would be the apparent source?

It seems like there should be an easy way to do this, but most of my searching turns up resources for resolving a hostname to a dynamic IP address, so I may not have the right terminology.

If it makes a difference I have a cable modem and use a router to set up a small internal network. Nothing fancy.