The web preview in iMessage is performed by each client. So your friend should see their IP in the preview since it's their client making the connection on their end.
This is easy to test if you've got another device with iMessage. Take your phone off WiFi and send yourself the link. The device on WiFi should show your home ISP address while the device on cellular should show the cellular provider IP.
That does not appear to be how iMessage works. I just tried this and my IP did appear on the receiving end. Since I edited the URL in the actual message it is possible the preview was cached from the original redirect.
Either way it is infuriating that iMessage does not show you what it is actually going to send.
This is easy to test if you've got another device with iMessage. Take your phone off WiFi and send yourself the link. The device on WiFi should show your home ISP address while the device on cellular should show the cellular provider IP.