I've had the weld-behind-the-blade thing happen in the past, on a motorcycle windscreen (which was expensive!). Hand tools are the way to go, and sand, don't attempt to plane.
Decades ago, I had a summer job in a light engineering factory. We made a lot of small parts - scales, verniers, etc. - from Perspex (= Plexiglas = acrylic). We always finished the edges with a slow-running linisher (belt sander). You can get a really good finish by ending the sanding with Jif-type cream cleaner, followed, if necessary, by toothpaste. There's also some special polish called Novus which you can find on eBay.
You can theoretically melt the edges to be smooth by running a blowlamp along, but use a butane one as it's cooler, and don't hesitate anywhere. I struggle to make this work well though. I'd guess you might try a domestic iron and greaseproof paper, but it might not be hot enough. There's a very fine line between melting and burning the stuff.
I've had equipment cases made up in 3mm and 6mm in the past (for exhibitions). I used an architectural modelmakers, who were happy to do it. They had a special Perspex bonding system with UV curing glue: you put a few drops along the joint, then hit it with a really strong UV source (with a fibre-optic light pipe, I think). The UV helped the glue spread and cured it, so the joint became almost invisible, and with no bubbles. Bubbles are almost impossible to avoid otherwise. the UV curing glue doesn't spread beyond the joint (usually) so the result looks really good, but isn't quite mechanically strong as the unjointed material.
I've recently seen other Perspex glue on eBay. The original formula was chips of perspex dissolved in Chloroform - highly inflammable, quite nasty to humans, and with very limited shelf life. I don't know what the current stuff is like, but I have to get some soon to repair a crack in a windscreen.
Hope you succeed, but I think it's one of those jobs where hand tools really win over power. As mentioned, a backing board may be a good idea too, to mimimise breakout.
Hope it goes well,
E.