Not sure, if this will work for all the pieces of LinuxMCE (it should, if all pieces are using the same config), but here is how to move the LinuxMCE MySQL DB to other host (from host1 which is Core to host2 which is your heavy machine): the DB that is used by LinuxMCE is specified in /etc/pluto.conf, so you can move DB and reconfigure LinuxMCE:
0) stop the inuxMCE if it is running
1) on host1 backup database 'pluto_main'
2) on host1 backup configuration file /etc/pluto.conf
3) copy backup of 'pluto_main' DB to host2 and restore DB there
4) edit the /etc/pluto.conf to change the values of MySqlHost, and username/password if they are different; verify that you can connect to DB on host2 it from host1
5) stop the mysql server on host1
6) start LinuxMCE and see if it still works
note that if you have extra M/D (media directors), you probably need to change their configuration file also
Don't know about Asterisk..