In reference to OP, as you refer to 737, InSoMnIaC is right:
You CAN NOT engage LVL CHANGE unless you get out of the APP mode by pressing TOGA!
Bad instructing done...
Maybe he needs to go into the manuals himself.
Yes, when not in APP mode you could in theory use LVL CHANGE if you have a missed approach altitude set, but this is just altering existing procedures.
Like other posters mentioned about different types, reduced Go Around thrust is initially set and will give sufficient climb gradient unless FULL GA is required with a second push of TOGA.