If there are moons with vaguely similar orbits to Earth's Moon, you can estimate their distances by:
  1. measuring parallax from widely separated observing points; or
  2. measuring parallax between different times of day (diurnal parallax); or
  3. measuring the size of your planet's shadow on a moon during a lunar eclipse

The latter two methods need your previously estimated diameter of the new earth.
Or you could use radar, if you are willing to use 20th century technology.