Hi CalNewbie,
Haven't seen the remake of the classic, but saw the original years ago. One thing I remember was the sheer numbers of zombies, (also if they were further than arm and teeth distance, they were not a threat, zombies weren't real smart

). And as much as I like a shotgun for close range defense, I'm afraid you'd be limited to carrying no more than 100-200 shells. And even that would not be light.
Of the current guns I own it would either be the Ruger 10/22 (a .22 to the brain is as effective as anything else) which would enable me to pack probably 1000 rounds. (If I took the 10/22 I'd probably carry a heavier centerfire pistol. Maybe a large capacity 9mm.) Or if the .22 were lacking, then my Ruger PC4GR would get the nod. It also having a ten round magazine chambered for .40 S&W would certainly do the job. Of course the other advantages being the ammo, while heavier than .22LR, is considerably lighter and less bulky than shotgun ammo. And an even bigger advantage is I'd also grab my Ruger P94 chambered for .40 S&W. The carbine and the pistol use the same magazines. A very big advantage when things get tense!
Whoever my "partner" was, probably my wife, would get an SKS with plenty of preloaded stripper clips. And whatever large capacity handgun they chose to grab. Maybe we'd even stuff a little Kel-Tec in the pocket. At 6.6 to 7.2 ounces (.32ACP or .380ACP, but both are 10 ounces loaded) they'd add very little to the carry weight.
Okay, okay, you said one! But you know how gun people are. One is never enough!

Besides, who'd enforce the "one rule"? I have a feeling BATFE would have other business on their minds!
Dave