Your math is correct on the volume, which is about 300 gallons.
I don't know where you live, but in most places with zoning restrictions a 3' deep pond must be fenced securely enough to protect children from getting to it. I bring it up only just in case you haven't checked on it.
Could you post a picture of the pond and it's setting so we could see your restrictions on filters?
Here's a piece of good news. The "head" is not the vertical distance from the pump in the bottom of the pond to the outflow of the filter. It is the distance from the surface of the water to the outflow of the filter. The pump really isn't lifting anything until the water is being lifted out of the pond. So you have 1 foot, not 4' of head.
Now comes the bad news.
The size of pond that a filter can handle according to the manufacturer is not actually a lie. It's just deceptive advertising. When they say a filter is for ponds up to 1000 gallons, they don't bother to add, "unless you have fish in the pond." Some of the more responsible companies will give a separate capacity for ponds with fish. My suggestion is that you divide the max by 10 to get its capacity for a pond with fish.
Your pond isn't big enough for koi. A good rule of thumb is that a koi pond should be at least 1000 gallons (to give swimming room for these big fish), and provide at least 300 gallons per koi. And that is with good filtration. I'm sure you've been told that fish will only grow enough to fit their container. To a certain extent that is true for goldfish. If they are kept in crowded conditions with little food for their first two years of life and survive that treatment, they will usually stay small. It is very difficult to keep koi small without killing them. Koi that don't reach average size (~ 30 inches) because of overcrowding will still reach 18-24 inches and produce as much poop as a pot-bellied pig of similar size.
So what are your options?
Stick with the limited filtration you propose, and stock very lightly. You could keep 6 goldfish healthy (with no other fish), but it will take a lot of cleaning and water changes to keep the water from getting ugly.
Hide a filter. Some people put the filter away from the pond, around a corner, behind a fence, in the garage, etc. You can run pipes on or under the ground from the pond to the filter and back. You can sink a filter into the ground, but that makes it difficult to clean. Many people make a waterfall in front of a large filter to completely hide it.
Make a filter pretty. A filter doesn't have to be in a barrel. You can get a large flowerpot or urn for your filter container and grow plants in the top of it. There are some beautiful rain barrels that make excellent filters.
If you have the ground space, you could make a bog filter. It looks like a lush garden next to your pond. I'm building one for my next-to-the-front-door pond.
