PoE is pointless unless you’re driving APs or VoIP phones, as Paul says. A PoE switch will provide power over Ethernet, not receive it. An AP was on your initial spec sheet but I don’t immediately see the point if you’ve got a mesh system.
If at all possible, I’d avoid making your own cables up and just buy patch cables of the required lengths. Crimping Ethernet cables is fiddly, there is a skill to it. I’ve done hundreds and I still don’t always get it right first time. If you’re intending running cable through drilled holes where the plugs would foul then I’d strongly suggest terminating with wall sockets rather than plugs, it’s a considerably easier job.
You don’t need a punch-down tool at all unless as above you’re planning on installing wall sockets, in which case investing in a better tool as someone else suggested will save you a lot of pain. So-called “disposable” punch-down tools are scarcely better than stuffing wires into terminals with a screwdriver.
I’ve used bargain-basement cable testers – I bought a handful for the apprentices at work once – and they were terrible. There’s little point in spending hundreds for a one-off job of course, but I’d swerve the absolute budget end. And again, you don’t need one at all unless you’re doing your own cable runs.