pcb design: how to route through hole connectors with tightly fitted pins?
No matter what you end up doing, you will probably end up needing to make 'escape routing' to the outside some way or other (multiple layers with a consistent escape strategy for example) and then route everything from there. If you're really dead-set on using autorouting, get it to that point and then let it have a shot at it. You will probably end up with a lot of vias.

Maybe something like this