Change the shift and passing gear linkage after noting their position (make sure it's in park!) The two trans should look exactly alike except the 727 is a little fatter. Same length, same bolt pattern.
Run the part number for the new converter to make sure it's correct for that trans. Lockup was used in early years of GW and 80-83 Cherokee/Wags and the converter was different. Lockup on top and non on bottom.
As for the flexplate, it depends what year Eagle you have and which 727. In '87 Eagles and FSJs went to the same starter as XJs and got rid of the PITA Ford style ones. At the same time they changed the offset of the flexplate 3/8" or 10mm closer to the motor as the XJ starter is shorter. They also went from 3 evenly spaced and 1 offset converter bolt to 4 even spaced ones. You can find out by matching the plate to the converter holes. Early flexplate has a dual pattern for the converter bolts (9XX and 727) Not sure on the later one, but one bolt won't line up. You can save $$ and drive/file the odd hole to fit but the flexplate from an '87 to? Wrangler 4.0L 999 will work and you can drill the 727 for the crank sensor while it's out to upgrade down the road.
Clear as mud? If you run the FSJ tcase, you have to bend the shifter away from the trans slightly to clear the low range linkage.