Knowing the router helps, thanks.
The Dewalt shows a 7A load on the sticker. Assuming you run a separate 20A circuit off your wall box, that leave you with ~10A plus a little overhead.
This depends, of course, on what your 60V PSU pulls, but I think you can get away with what I suggested earlier - having a relay-controlled 120v socket on the CNC control box to power the router motor.
The result would be a very handy/tidy setup. Single power cord from CNC box to a dedicated 20A receptacle, and a receptacle on the side of the CNC box to power the router motor. It would certainly be user-friendly to have LCNC be able to turn on/off the router.
And having a separate circuit for the CNC means if you start up a table saw or some other big load that won't trip your CNC unless the offending equipment pulls your whole house main breaker down.