Honestly i think your best bet is going to be to not print directly on the brown paper. I think the better solution would be to print on clear plastic labels, and print white text, and then peel off the labels and stick them onto the invitations. I don't think it's easy to just go buy a white ink cartridge, and if you can, then you might have to have a printer for which they actually make white ink cartridges. but EVEN then, you're color and contrast is going to be a hundred times better if you print on clear plastic, and then stick the clear plastic to the brown paper.
...Let me know if you have questions! :-)