You say there's no room for a connector on the edge but the track going to LED8 could easily be moved and you could have the pads on the side of the board. It doesn't appear the track layout will have any impact.

I'd just have 3 0.1" spaced pads on layer 2 ala an old PC ISA card and then make a 3 way connector to "plug" the boards into for programming.

Or have the boards already been made? If so, none of that helps