I started to disentangle my layout in between gooning sessions like >>2951838 suggested, and it doesn't seem quite so bad. I think I might even try to do it surface-mount, even though I've only ever done through-hole stuff on stripboard before now. Does SMT actually make routing easier? I guess there are no features poking through the board to disrupt routing on the other side. I wonder if it might be too ambitious for my first SMT attempt, but I also don't really want to spend time making a "toy" practice project first.....