Hey guys, first time posting. I had the same issue with the chief engineer/rocket bug (not the dieing and being stuck part). I put the ramjet upgrade on my hot bar and just ran around the room spamming the button and jumping on things. If you're facing the engineer from the floor, I jumped on the computer banks to your left and noticed the rocket tried to appear on my back for a moment. Then the group agro'ed me and began firing. I jumped back on the computer bank and spammed the rocket while jumping for about 3 seconds until it finally activated.
Long story short, run around the room (nearish the platform but get further while you experiment) spamming the ramjet and you might find the sweet spot to get launched up.
No guarantees but hope it helps!
Thank you soo much, i did exactly what you said and it worked, computer banks on the left guys.