The OP wants to wear the tyres (part-way) out on one edge first, then move that edge of the tyre from the inside to the outside in order to then wear the other side part-way out while the original worn side gets a break while on the outside. This means the tyres have to be removed and refitted, even if the wheels are not swopped around.
If the OP specifically wants non-directional tyres to do this exercise with, the point is that he will have to find ones that are symmetrical. These tyres in large sizes are often only to be found from poor quality budget brands.
For the OP's purposes, tyres which are asymmetrical will not work at all. This is because asymmetrical tyres have a specific inside and outside, as DavidC said. So when switching the tyre from left to right, the same side of the tyre still has to be on the inside. If the OP is experiencing inside edge wear, this will therefore not make any difference to the premature wear on the inside of the tyre.
Even with a symmetrical non-directional tyre, the OP would still have to remove the tyres from the rims and refit them with the other edges on the inside, otherwise the same would happen. The same edge of the tyre would stay on the inside of the rim if you just swop the tyres+rims without removing the tyres.
To visualise this, take four coffee mugs and put them down on their sides. Imagine the open ends are the defined inside of the asymmetrical tyres and the closed ends are the defined outsides. Now swop them across the car. You can't now end up with the closed ends facing inward. So the same ends end up "inside" again - the excessive wear will still be on the same part of the tyre.
To move the excessive wear to the other edge of the tyre, they have to be removed from the wheels and refitted the other way around. With a non-directional, symmetrical tyre, this can be done and each wheel just be fitted to the same corner of the car again. With a directional tyre, once the tyres have been turned around, the wheels that used to be on the left need to go to the right and vice-versa, otherwise the tyres will be rotating in the wrong directions.
So, my point is that the easiest thing to go for would probably be to get directional tyres. These are common enough from high-quality manufacturers and can be found in sizes to fit the OP's car. Still, to spread wear evenly across the tyres by swopping them from left to right, the tyres will have to be removed from the rims and refitted.
This is the only way to achieve what the OP wants to achieve. (Apart from getting the alignment done properly and fixing any other problems which may be causing excessive inside edge wear.)