Thanks for the detailed reply. I'd assumed your source files were digital photos of artwork (e.g. 20 megapixel). I want to print some photos of Japanese wood block artwork, and there's an RNN implementation which apparently works well for doing 2x, 4x etc. upscaling: https://github.com/nagadomi/waifu2x
Ah, yes photos are difficult. I have used some of the pixel art upscalers, but not for prints, and I haven't used waifu2x before. They can be almost magical in some situations, and Japanese woodblocks seem like a pretty good application for waifu2x.
I'd worry a little about textured background and what happens in areas of high detail, but I'm speculating. You can see resolution artifacts in the waifu2x examples. But those source images are tiny -- it may behave completely differently on a 20MP image. It also depends on what material you print onto. Canvas and some popular art papers have a lot of texture that can hide the 1-2 pixel sized artifacts.
If you are taking the photos yourself, one option you might consider is to take overlapping photos of sub-portions of the woodblocks and stitch them together using a panorama editor like Hugin to get a source image that is at or above your target resolution.
If you have no other options but upscaling, do some experiments with the cheapest printing service you can find that will print 24x36 inches at 300dpi. There are a bunch online that will mail you prints very fast. If you're in the US, Costco does ridiculously cheap photo prints at 24x36. Experiment with upscalers and see what works best for the photos you have. You might find that you don't like any of them, or you might find out that you can't tell much difference between waifu2x and generic upscaling, or you might find out that waifu2x is perfect and unbeatable.
