The way I understand it, the Adobe RGB in the Print module applies only to the previews, and draft printing, but I may be wrong.
See Lightroom Help | Color management. Under "How Lightroom manages color" it says:
"Lightroom uses Adobe RGB:
- for previews in the Library, Map, Book, Slideshow, Print, and Web modules
- when printing in Draft mode"
- ..."
In another help document Lightroom Help | Print job options and settings it says:
"To send the image data to the printer driver without first converting the image according to a profile, choose Managed By Printer."
So, I assume LR is sending the image in its internal color space, but then again it is just my interpretation, although it is compatible with what happens, i.e. the "color numbers" from a big color space are sent to a smaller color space, without conversion, and the colors get washed out. It is much the same as when an Adobe RGB encoded image is viewed on non-color managed program.
As you do not have specific profile for your printer and get good results using Adobe RGB you may want to try to cheat a little.
Try this: in the Color Management tab of the Print module, instead of selecting "Managed by printer" select "Other..." and you will see a list of the available profiles on your computer (eventually ticking "Include display profiles"). There, choose Adobe RGB.
In any case, I'm convinced this is NOT the correct approach.