There is a scripted checklist for an engine fire. It wasn't dreamt up overnight.
If Boeing felt there was value in running a system test during that checklist, then they would say so. They don't. In fact, they say the opposite in the QRH introduction.
Prior to running a system test, good CRM directs that you first explain to the other pilot what you are about to do, and not do it until he/she is comfortable with it. Since you are going outside the checklist, good CRM directs that the other pilot will very likely question your actions and you will have to explain further. "Why are you conducting troubleshooting - we haven't even finished the checklist yet." The will result in distraction from the emergency at hand, cause possible delay to continuing the checklist and landing the aircraft, and (worst case) may result in you performing actions adverse to the continued safety of the aircraft due to conducting un-rehearsed and non-recommended actions to the amazement of the other pilot.