Supposedly the Mac Pro Firmware Restoration CD only works if the Mac Pro firmware is 'broken'. In your case as the Mac Pro boots and runs this is not true. It will only cost a blank CD to try though.
For what it's worth I have been able to installed Mojave 10.14.0 on my Mac Pro and that has resulted in my boot rom version being completely messed up at least as displayed. It shows only 138.0.0.0 or 138.0.0.0.0 depending on where you look. It also works even with APFS on an SSD drive so it is apparently a cosmetic issue.
A more complex approach you could try is that someone previously hacked up an installer to upgrade MacPro4,1 models to MacPro5,1 firmware. This same installer can also downgrade MacPro5,1 firmware back to MacPro4,1 firmware and you could then try upgrading again. This only works on 4-core models of Mac Pro which means it should work on yours, it will not work on my 6-core model.
Personally I would not risk doing this as it is working.
With regards to upgrading to Mojave can you answer the following.
What other video cards do you have?
What type of drive are you trying to install Mojave on to? How is it connected?