It is a while since I had a Lhires but I would check that the slit is in focus of the doublet.. It is too easy to get it out of focus when focusing the camera. You will need to do this on the bench with the grating removed.
I don’t see how moving g the mirror will have much effect..
I would be surprised if it was the main mirror that needed adjusting, but it is an easy adjustment to try. There is a screw at the bottom of the LHires III that you can loosen to allow a small movement of the main mirror. The user guide has instructions on how to do this including pictures of the procedure.
Was it definitely the main mirror and not the guide mirror? The user manual also has instructions on how to adjust the guide mirror.
That said, my first place to start would not be the mirrors, but the main collimator lens and guide camera focus, unless you have already tried these adjustments.
You can leave the spectrograph on the telescope to adjust the collimating lens (doublet) focus, in fact this is normal practice. You access the lens via the little doors at the side of the spectrograph. I expect you know how to do this and the easiest way is to focus it using the spectrum of the neon calibration lamp.
I’ve found that the sweet spot for my Lhires III does not have a perfectly thin spectrum height, though others may have tweaked theirs to give better results. There was a long discussion on this topic in the below forum thread.