You can run the script with screen
or nohup
but I'm not sure how that would help since the script does not appear to log its output to any file.
nohup bash script.sh </dev/null >/dev/null 2>&1 &
Or
screen -dm bash script.sh </dev/null >/dev/null 2>&1 &
Disown could also apply:
bash script.sh </dev/null >/dev/null 2>&1 & disown
You should just test which one would not allow the command to suspend or hang up when the terminal exits.
If you want to log the output to a file, you can try these versions:
nohup bash script.sh </dev/null >/path/to/logfile 2>&1 &
screen -dm bash script.sh </dev/null >/path/to/logfile 2>&1 &
bash script.sh </dev/null >/path/to/logfile 2>&1 & disown