As Garbz says, you'll have problems getting better reds than the 2400 can produce - it's actually one of the better printers when it comes to reds. Even the Canon Pro9500, which has red ink, doesn't do much better (in fact it doesn't do as well as the 2400 with the brighter reds). Printing on light-sensitive paper with a Lightjet or similar will be even worse. The following graph assumes that your images use sRGB (of course they could be in an even larger space, and hence even more out of gamut). This is a slice through the image gamut (the second image you posted, values shown as dots), the gamut of an R2400 with glossy paper and the gamut of Fuji glossy paper printed on a LightJet, all at L=60. The outer extents of the print gamuts for all values of L are also shown - the R2400 is the outside line for most hues.
If you really want to pursue this you could ask someone like
Lenny Eiger who has a 12-color Roland d'Vinci printer, I think. I don't have a profile for that, so I can't say exactly how much better it will be than a 2400, but it will be about as good as you can get (and it won't be cheap).