One thought process, I have gotten away from the incremental process, for me takes too long and I don't see much gain.
Software to consider would be Blow up, Perfect Resize (former GF) and Photozoom Pro. I would lead with the later and use the Spline 64 as the uprez algorithm and look at Lanzcos. My method is to get the image sharp before the uprez, then use Focus Magic after the uprez, if needed.
I print everything now from LR, and for Epson you need to use 360 or 720ppi for the best results. For this size work, I would stay with 360ppi.
You can also simply try allowing LR to uprez the image for you if you are within 180 to 240ppi, and apply some output sharpening. LR does a very good job getting out the final image and often times, I can't see much difference between a 240dpi shot uprez'd to 360 with one of the software tools I have referenced and printed in LR at 360ppi and using LR to print a file from 240dpi, as 360ppi output with respective output sharpening.
This works for me and my customers but I am very sure you will get a lot of other ideas.
Paul