Ok, I may have spoke too soon.
But, I have learned more!
The DMZPlus configuration with the UVerse modem wasn't agreeing with my device. I guess the one test we didn't do was see what happens by basically double-NATing. I'm currently doing a double-NAT [WAN address of my router is a 192.168... address, and LAN is on a different subnet], but I haven't fully tested to see if there's any additional oddities that are happening from it.
The weird thing is the DMZPlus configuration works wonderfully with my DI-524. It was up for the almost week while I was cross-shipping.
My best guess is there's something the DIR-651 doesn't like with the UVerse DMZPlus.
So, armed with this information, I may spend the next couple days determining one of the following:
1) There's an extra configuration setting missing between the DIR-651 and UVerse modem with DMZPlus or
2) The DIR-651 doesn't like my UVerse modem's DMZPlus at all and therefore is not the best router for me
So, the short is getting a 192.168... IP address with my DIR-651 to my UVerse modem keeps the connection stable [an hour and a half thus far], but getting the public IP via DMZPlus doesn't let the connection stay up for longer than 5 minutes.
I'll see if I can get more information posted after additional tests.
I don't know if it explains why my device wasn't letting me stay in [video I posted above], but I have more information!