I've just bought a telescopic spinning rod to take touring with me and I'm going to get a 7 or 8 piece fly rod next year so I understand the need to fish and ride.
If you want to ride on the beach get a fat bike no question. A Krampus or Ogre will stop rolling [with you on it] as soon as you hit sand. You might get away with some riding on really compacted sand, but that's not something I would count on.
So your real choice is 4" or 5" tires. You can make 4" tires work and they'd be better on the road, but 5" tires will let you ride more on the beach so you need to decide what's most important. Factor in how much you weigh and how much gear you'll have with you to determine if you need max floatation.
If you go 4" tires get Black Fyoyds or Larrys. They will roll well on pavement and sand. 4-8psi on the beach and 15-17psi on the road. A Pugs can rock 4" tires nicely.
If you go 5" tires get Big Fat Larrys. Similar pressures to above....a little less on the road pressure. A Pugs can fit 5" tires, but you may have some chain rub in the lowest gears. A Moonlander or the new Ice Cream Truck will swallow 5" rubber with no issues.
Have an awesome time. Tight lines!
Or get a duel purpose rod...
Eagle Claw Trailmaster Pack Rods