It should work. If there are significant firmware differences some syntax may have changed.
After loading the old configuration onto the 1534p you can run the show output-errors command to see if anything couldn't be parsed and if so manually edit the configuration.
Yes, you can indeed upgrade to the 1534 with no trouble.
Adtrans AOS should be able to handle it since they are of the same family of devices.
The 1531 has 10 gig eth ports and 2 fiber ports (port 11 and 12).
The 1534 has 24 gig eth ports and 4 fiber ports (port 25,26,27,28).
Just do a 'range' on your interface builds (if they are built identical and range gig eth ports 1-24 for the 1534 device.
You can drop the current config of the 1531 into the 1534, then add gig ports 10-24 and 25-28.
Should work well for you, I deal with them daily.