I am not sure what I can do to fix this issue, but as I posted below, I just cannot get Remote Play to work correctly. I just tired a few more different ways to get this setup, with no luck. At this point I have to assume that there is some sort of incompatibility between my ISP, my router, and my PS3. I guess it could be user error, but I have no idea what to try next. Very frustrating!
If I enable Remote Start, the PS3 will turn on every couple of minutes. I know that if a PSP does not connect within a limited time window, the PS3 will power down. At this point, I will not be able to get the PSP to connect via Remote Play. I keep turning the PS3 off (press the O button to return the XMB and then power down via the controller), but it will power on again within two minutes.