Well not sure which version you are running on. I'm on 11.x I dont see mv option to rename objects. Guess yours is 12.x or above. The issue you are seeing is pretty common when large content is pasted. So inputs gets jumbled. Like tmsh alone gets pasted and the rest commands again getting pasted. In your method 1, the tmsh was pasted, and again inside the tmos you cannot rerun tmsh command. hence the syntax error.
To avoid this use transaction.
tmsh
create cli transaction
mv ltm node /mypart/10.1.1.1 /mypart/newname1
mv ltm node /mypart/10.1.1.2 /mypart/newname2
mv ltm node /mypart/10.1.1.3 /mypart/newname3
mv ltm node /mypart/10.1.1.4 /mypart/newname4
submit cli transaction
Or you can paste 5 lines each in the bash itself and it should work smoothly. Then slowly increase your count to 10.