That battery should work fine. Though you don't need a variable voltage battery. I ran my minime on 9volts. It is unfortunate that the cheap wally world batteries are no longer available (I need a new one). One of those for $30-$40 would be ideal.
If you buy used, you need to watch out for shelf life issues (lithium degrades even when not being used). Mine is on its last legs.
Fwiw, your friend's comments are somewhat off base. Lithiums are wayy more dangerous than SLAs (except to your back!).