Since you're not going the router......um route,

here's what I suggest.
Take some sandpaper (around 300 grit) and wrap it around a sanding block. Use this to knock the sharp edge off the lexan. Be careful that you don't drag the sandpaper over the lexan face and scratch it. That should be enough to take the super sharp edge off it. If you want to polish the edge you can move up to maybe a 600 grade sandpaper and THEN a polishing compound. If you don't want to buy a polishing compound, toothpaste will even work. I think sandpaper is probably enough for your purposes though.
Now the edge. Now that the edge of the lexan isn't sharp I wouldn't try to cover it as well. How thick is your board? 5/8?
Two choices:
1. Either sand the edge really nice and smooth and paint it black.
2. Find a kind soul here who has some left over 5/8" wide t-molding to donate. (I probably have some) Cut off the "t" part of the molding and glue it onto the edge with hot glue. In your application it will be held on plenty strong and to be honest you really can't beat the look of t-molding.
You mentioned perviously that you may use black electrical tape for your edge. I suggest you don't. Have you even seen what old electrical tape morphs into? The adhesive melts and turns into a sticky, gooey mess. If you don't want to go with some sort of edging then I'd personally do paint.
But...... as you see there's lots of choices. Make it your own! There's no "right" way to do it.
....except MY way of course! *laughs*