DE gives you the best filtration, Cartridge the next best and Sand the least best.
But, from a performance standpoint they all perform equally well at removing particulate matter from your pool. Unless you spend time staring closely in your pool at night with the lights on you will never know the difference. You just can't see it.
The downside to DE filters is that if you have serious algae problems DE filters clog faster and take more time to clean. But you are on TFP and will follow the TFP method and won't have algae problems right !!!! So it doesn't matter to you.
The secret to a trouble free filter is to grossly oversize the filter. Purchase a filter at least twice the recommended size for your pool. That way your pool operates at lower pressure, and like Matt above you only have to clean the filter once or twice a year. I clean my cartridge filter twice a year.
Just my two cents, if your existing walls are level and in good shape I would leave then and redo everything else. Concrete walls generally are better at staying in place. In other words they don't tend to move as much in the winter. The less you disturb the soil then the less you are going to worry about compaction.
Right, I'm not going to make my filter choice based on algae issues, as there shouldn't be any. It's good to know there's not a huge difference in water clarity from 4u to 10u.
I don't think I have many options to grosssly oversize the filter, since the pool will be 40k-50k gallons.