Look out, long answer below... You've kept a pond before, so I apologize if you know most of this already.
As others have mentioned, the pond is a bit shallow and may heat up more than you'd like in the summer and freeze solid in the winter. But if you take the goldfish inside and don't have things like hardy water lilies that will die if they freeze, maybe it won't be a huge issue for you.
If you wind up with Koi or otherwise dramatically increase the biological load of your pond, you would certainly benefit from a bottom drain and the more advanced types of filters (trickle towers, moving bed, etc.) If not you have a lot bigger range of choices that will probably work for you and really the main thing is to find how how to balance your pond - and your expectations for water clarity.
Regardless of filtration, in your case one of the biggest things is that your pond is new and hasn't cycled yet. Eventually, the liner, any rocks in or touching the water and even the surface of pots and any plants you add can provide a surface for beneficial bacteria to grow which will help at some level. And then any plants can use the remaining nutrients. For example, I have some container water gardens that have never turned green but were started with plants in containers that came from the main pond. They aren't crystal clear, but I can see the bottom. Also, when you add filtration it won't have cycled either so it may not show the immediate benefit you might expect. Just stuff to keep in mind.
If you add the UV, it will kill the algae, but some would suggest that can even slow down the cycling. I have crystal clear water and have never used UV. I'd like to believe that if you get the balance right and keep your number of fish low enough you might not need the UV, especially once some string algae or other types show up that some have suggested will actually inhibit the algae that makes your water green. But I don't have full day sun like you do so I don't know if UV would always be essential for you or not.
So here is what is currently working for me that might possibly work for you although you'd need to scale it up. It is a pretty simple and quite inexpensive. I have a 350-400 or so gallon pond (maybe 450 if you include the bog). My filtration starts with the TetraPond SF1 you mentioned. I have it attached to a 600gph Tetra Pond pump. The pump pulls water through the SF1 (which I added some rocks to beneath the filter pads that came with it) then is pumped into a small bog where it runs through the pea gravel then back into the main pond over a small waterfall. The SF1 is basically a mechanical pre-filter to keep larger particulate from getting into the bog, but the pads and few stones also provide some surface area for bio filtration as well. In the bog the water goes through the pea gravel, through the roots of my bog plants in the pea gravel (a couple graceful cattails, some pennywort, pickerel rush, a couple iris) and then into an area at the front of the bog where the pea gravel is lower (about 4 inches below the water) and I keep a few water hyacinth floating there. How effective is any particular plant and how much do the roots do vs. just the surface area of the pea gravel the water flows over or the floating water hyacinth? I could guess, but really don't know. All I know is that the balance of all of it works. By the way, I have 4 goldfish (almost 5 inches long now) and about 20 Rosy reds. In terms of maintenance, I've only cleaned the SF1 out once the entire year and just rinsed the pads in a bucket of pond water, since using my tap water would kill the bacteria in the filter pads. Personally, I only clean filters if I see restricted water flow or have another reason to think they really need attention. In this case, I had a lot of suspended clay when I first started the pond and rain washed runoff over a low edge so I knew there was a lot of clay built up in the filter. Otherwise I might not have cleaned it out all season. I've had aquariums with filters I cleaned only every 6 months and the water was clear and the fish lived happily for years.
If you wanted to consider a bog area, I think running through 2 of the SF1 filters would be a good start and they are very inexpensive. Then, if you are willing to give up some of the open water space, you could use some stones to build up a small wall sectioning off part of your pond with the black spray foam to fill voids at the lower area, and create a bog area with pea gravel behind the wall. Either leave some of the openings in the stone wall open to allow water to flow through or leave a section lower to let the water tumble over it. Or maybe there is an area you could create behind your waterfall with a small liner? If you choose hardy plants for the bog they should survive freezing just fine.
If you just want to go with the simplest thing and try to get away without a bog or external filter, you might try running 4 of the SF1 filters for a while and see how that goes. If you wind up with more murkiness than you'd like, you could add some quilt batting to a couple of these to help pull out the smaller particles, but you'd need to clean those more often. There are certainly more powerful filtering options out there e, but using the SF1 filters would let you keep it all in the pond and not be something you're trying to hide outside the pond. I put one of my lily pots on top of the filter so it helps raise the plant up and hides the filter at the same time, so with your shallower pond if you add plants that like being closer to the top you could do this as well, but just make sure not to block too much of the inflow.
What you decide on for filtration is really about achieving a balance in your pond and depends a lot on your needs/expectations. Do you need/expect crystal clear pond or do you just want it relatively clear so you can enjoy the fish and not be grossed out by looking at your pond? Plants really have to be part of your equation when choosing filtration since without any plants you will need more filtration. You may already have it in the plan, but adding a few water lilies or other plants that give you partial surface coverage will definitely help in both cutting down how much light hits the water (and grows the algae) as well as helping to keep your relatively shallow pond from over heating. This is probably even more of an issue for you next summer. Due to the depth of your pond you could either try to over winter the lilies in a cool basement or even just grow them as annuals if you don't want the fuss. If you choose tropical water lilies you can get them to set tubers in the fall and store those pretty easily to restart in the Spring.
Floating plants like water hyacinth and water lettuce will help dramatically as well by using up nutrients that otherwise just feed your algae. As an example of what they can accomplish, I have a neighbor with a 1000+ gallon pond with 3 water lilies and 20+ larger goldfish - possibly a lot more little ones that are still dark and hard to see. All she has ever run was a pump with a simple sponge pre-filter running which had very little effect besides just moving the water around. It was very hard to see her fish due to algae and particulates in the water and and then the pump died and by late summer her pond was even greener and you couldn't see more than an inch below the surface. I gave her some water hyacinth, they multiplied (at least a dozen big ones in there now) and now you can see at least a foot down into the water and the water only has a slightly green tint. With no mechanical filtration the little particles in the water making it less than crystal clear will always be there. The plants were the only addition. The fish are easy to see and her lilies are blooming so for her this is clean enough and frankly looks like many natural small ponds. Most people would tell you that you can't have goldfish pond without a filter, but she is clearly a filter atheist. She's been admiring my bog filter and will probably add one next year, but only because she wants to grow all the plants. The added filtration isn't that important to her, although maybe she will see the light when she can see the bottom of her pond.
Hope some of that helps.
