hey guys! Great to hear from you.
I haven't done too much cutting with knives but we've been test-cutting in my club with swords down to machete size (including cheap machetes) all the way up to montante size, as well as spears, axes and a few other weapons. I have an Albion sharp and a handful of cheaper ones, as well as two antique swords. in the club as a whole we have maybe 10 real swords (sharps) and 10 or 15 cheaper things like machetes and so on that we still like to cut with.
So I'm not sure if what I'm going to describe is ideal for your purposes or not, just keep in mind the caveat that this is to do with test-cutting with swords not really knives.
Reasons to test cutThink about what you want to get out of it. The swinging target sounds interesting as a way to practice cutting at (or stabbing) a dynamic target. Usually in the HEMA scene though we emphasize learning to cut first, then go up into dynamic targets (like tatami mats on a rolling base for example) once you learn the basics. I'm still at the point of learning the basics really, maybe to an intermediate level, but I've mostly cut at static targets. For me to learn accuracy I'd rather use a blunt since it's safer. Cutting itself is actually kind of hard to do well or right. It takes some practice at the very least. It's also fun to learn, a good challenge. Using both cheap and more expensive cutting media can teach you how to cut. This to me is enough of a challenge that you could use up 10 or 15 cutting sessions just getting up to speed ,though you may learn faster than I did.
Preparation for cutting and safetyThis may not be as important for stabbing things, but to cut well, you really need to sharpen your blade. We usually sharpen them a lot. There is a noticeable difference between sharp and dull swords. The sharper it is though, the more dangerous it is. I try to explain to people that it's like a "short range gun". Wear boots or heavy shoes, wear long pants with relatively thick cloth (like blue jeans). I have seen people get hurt at test-cutting events, including people who should know better. Pay attention to the blade, and pay attention to the people around you. Watch the point etc.
Cheap cutting materialsOk now the fun part.
Plastic Bottles. The cheapest and most forgiving cutting material for me is the humble plastic milk bottle. 1 gallon size in the US is best. You can put these on a post about chest-high, fill it full of water and cut. If you cut really badly, it will bounce away with no effect. If you cut 'ok' you will hack off a hunk of it and knock it over. If you cut reasonably well (and it's pretty easy with this medium) it will shear off the top and leave the bottom still full of water. This is very satisfying and fun. Also pretty easy. They are also a good training tool since you can look at your cut in the plastic and see if you scooped it, or stuttered, or went off your cutting angle or whatever, it shows up in the shape of the cut part.

Smaller water bottles require a little bit more finesse, but so long as the plastic isn't too hard or thick they are pretty easy. Thicker plastic obviously more of a challenge. (You can also bend swords, especially fragile or poorly made ones, even cutting things like plastic bottles so be careful.)

Soda bottles are a bit harder still. Fill it full of water and cut, it requires a 'decent' cut or else they just bounce away with a dent. Start with the milk jug and then move to these. Still cheap and still pretty easy to cut. But harder, and if you are pretty good you can 'carrot' these - i.e. cut them 5 or 6 times until just the bottom is left with water still in it. Very satisfying.
Pool noodles. These are also cheap, can be a bit tricky, bounce away with a bad cut but. Fairly satisfying to cut.
Cardboard tubes These are harder to cut, which can be satisfying. Obviously depends on the thickness of the tube, 3" or 4" wide is about right. It gets harder from there. Blunts your blade quicker and some types which have bits of metal wire in them can score your blade up - so it depends if you have a fancy blade or a cheap machete.
Bamboo if you live near some, this is also a pretty challenging and fun media to cut.
Game or animal cadavers This is a little icky, and might freak some people out, but I think it's harmless. We have done it many times. You can get meat and bones from your butcher, or from someone who hunts, and prior to having a barbque, hang it up on a rope and see how well you can cut it. Ranges from surprisingly easy to quite hard. Keep in mind bones that have been dried out can be a lot harder than in a living animal. Obviously never cut a living animal! Just being obvious for legal reasons here.
Pro-mediaBasically, tatami mats are the thing. You can get real ones in the mail for about $20, but you can also buy cheap as dirt straw mats from all over the place (in the US they sell them as beach mats) and to me they are about the same. You can also either put a wooden dowel or rod in the middle or not at your discretion. Thickness also varies a lot and matters a great deal vis a vis difficulty. A 3" tatami is about the same as a 2 liter soda bottle for difficulty. A 7" is much harder. You can buy a tatami mat stand for about $20 too or you can just make your own. A Christmas tree stand works pretty well too and lots of free ones are around after Christmas. Prep for the mats is fairly easy, just soak them in water for 24 hours before you cut.
Cutting tatami is a little bit hard at first. It's similar to shooting when you first learn to shoot - you have to train yourself to breathe, not to jerk your hand at the last minute, not to flinch and so on. But once you learn you can cut through all of them pretty easily with a good sword. My little constable will go through 3 7' tatami mats in one good cut. This is what you hone your skills on for the cutting tournaments etc. Start with easy cuts (oberhau) and then practice multiple cuts, true edge / false edge / true edge again. Cut it 3 or 4 times before the first piece hits the ground.
Cutting is tons of fun, especially with a few friends. I haven't done it in too long.
G