I agree with jayvee about addressing the issue in the cover letter -- particularly as the position is within the company, so the hiring manager will likely read what you have to say.
As for nationality and the like, I'd advise against leaving it off if you're already here. On the first couple covering letters I sent to potential jobs, I included something to the effect of, "I am present and settled in London, with full entry clearance to work on a spousal visa." Didn't even get a "ta, have a nice day" (and I wasn't applying above my station). Once I cut that, I got an email or phone call back on six jobs.
The first question out of agents' mouths was about my residency/work status (due to exclusively U.S. jobs on the CV), but none were put off when they heard it directly from me. The same factor also came up in the three first-round interviews I pursued and, again, no issue. In general (though not in this specific case), it might be best to apply in a way that demonstrates you have the confidence you're legal to work, and that doesn't apologise for/call attention to your "unusual" status.
I fully acknowledge my experience might have been coincidence, but I'm not exactly an unhireable candidate -- in the end, I had one offer, one company beg me to not drop out of the process, and another discuss creating a new position for me -- and I don't see another variable that would have turned off the initial round of companies.