How many cc engine do I need?

I'm new to motorcycles. I'm 6'4 and weigh 350 pounds. My Girlfriend weighs around 150. We will normally be riding it together so I need a motorcycle that can easily handle 500 pounds on it at highway speeds. I'm interested in the Honda VTX 1300. Would that be enough? Could I go with a lower cc engine and still handle 70 mph easily enough with 500 pounds?

Added (1). Would a 750 cc engine be enough? Also considering a shadow aero that I'm looking at. I like the Honda's and I hear they are reliable.

Added (2). I'm looking to spend as little as possible because I do not have much money to spend right now, but other good motorcycle recommendations are welcome.

Read, learn, and decide for yourself:
https://www.motorcyclecruiser.com/getting-loaded-how-much-weight-can-your-motorcycle-handle

Guess you are a new rider if you need to ask so a few suggestions-
Take a MSF Basic Riders Course before you go any farther. The BRC will have small basic bikes to learn on but your size and the bike's size are not that important right now, learning to ride is.

Ride whatever bike you choose alone for awhile than think about hauling somebody with you.

After you have taken the BRC and have an endorsement go shopping (it's FUN) to find bikes you are comfortable while not riding, just pushing around (first drops are in the garage/driveway) than decide.

Any V-twin or V-4 over 800cc should power you/Girlfriend around at moderate speeds BUT the bigger the bike means the heavier bike.

Also- taking the MSF BRC should reduce insurance costs for a begineer rider, an added plus. The course costs around $250-300 and worth the money in the long run (might even save your but by knowing right from wrong on a motorcycle).

Test drive a few.

If you're a big guy, then the amount of room on the bike is probably as important as the motor size. A smaller engine will cope but larger bikes are rated to carry more weight and have more physical room for passengers, so I'd say bigger is better.

On may bikes a rider and passenger combination of 500 pounds will cause it to exceed the allowable weight limits. You may well be looking at a very small range of large bikes to be able to handle that - it will need to be roomy too.

1300 would be more than enough.

Have you seen Youtube videos from with whole families on a Honda 50?

You're going to have to modify something… But try a Boss Hoss, dude.

350 is heavy. You should actually consider a car. A 500CC could haul the weight around as it is just rolling forward… It is the braking that is not up to the task. So little tire contact with the road. A car is at least 4 tires on the road. You stop faster. Remember stopping matters more than acceleration… Otherwise you EAT TAILGATES. But a 750 does too. The higher in engine size you go the heavier the bike becomes as well.

You definitely need bigger than 750! My nephew bought a 750 spirit as his first and his manual says the cap is about 400. Even if you and your lady didn't exceed the cap the ride would be extremely uncomfortable! Go big or go home dude, but be safe if you go over 1000. They got a lot of pickup