Ubuntu – Rename command not found when running a script

for f in IM*.BMP; do
    n1=${f:2:1}
    n2=${f:3:3}
    new="cam$(( n1 + 1 ))-$(( 10#$n2 )).BMP"
    echo mv "$f" "$new"
done

$(( ... )) is an arithmetic expansion. The 10#$n2 forces the contents of the n2 variable to be handled as a base-10 number, and the leading zeroes are dropped: you cannot use $(( n2 + 0 )) because you'd get an "invalid octal" error for n2 == 008. The $ is optional inside an arithmetic expression, unless you use the base# notation

PATH="/var/log/servers/Game/"

This sets the PATH to just that directory, and I doubt that directory contains date.

Be careful when you use uppercase variable names, you could easily be stomping over a standard environment variable.

Stick to lowercase, and be more descriptive. Consider using, for example:

log_path="/var/log/servers/Game/"
NEWFILE="${log_path}${FILENAME}_${DATE}${EXT}"