Some further ideas –
1) The surface skimmer is essential. Good idea. It is one thing that keeps the water nice and clear.
2) You already have some ideas as you say you cannot go with a gravity fed system. Therefore moving to a pressurised filter would be the way to go. On 3,000 litres one of the simplest filters to create is using a swimming pool sand filter and a pump that can drive it.
3) You would use 3 – 5mm gravel inside the filter. The fine sand in normal swimming pool filters clogs too quickly. If you can get hold of a Collins P24 sand filter it would work just fine. The Collins sand filters have port hole on the side which is easily opened for maintenance of the gravel bed.
4) You would need a small swimming pool pump – something like a Speck Magic 8. This would be strong enough to suck the water off the floor and through the skimmer and strong enough to push the water through the pressurised chamber then back to the pond via the waterfall and / or stream.
5) A Magic 8 would also match up nicely with the turnover rates you need.
6) Submersible pumps are not strong enough to run with pressurised sand filters.
7) One P24 Collins sand filter would work just fine on 3000 litres. And depending on the situation in the garden you may not need a UV light. If you do use a UV light you must place it after the sand filter so it is not under pressure.
If you do run a single pump for surface skimmer and bottom drain then you need to place a valve on the suction from the surface skimmer to control the flow between the bottom suction and the skimmer.
9) A few diagrams with the pipe work, pump and filter as well as possible placement places is needed.
10) And you are learning to keep water – which is great – then you keep koi. You just break everything down and build it on a bigger scale.
11) You mentioned making a barrel filter. This will work. Use it as a trickle filter and perhaps have some mechanical filtration in a separate container. Pump the water to the top and spray it downwards through the media. Exit at the bottom through pipes of sufficient diameter so the water does not build up inside the barrel. There will be no pressure inside the barrel but a lot of oxygen. You can use plastic shaving for a filter media.
12) You need to use swimming pool filters if you and going to close the filter chamber. Pumps can create quite a lot of pressure especially the swimming pool types – and a barrel / container which is not reinforced to take pressure will burst sooner or later.