You have a two part solution, but there are more parts to your problem.
1. The adiabatic expansion of an ideal gas will give you the temperature of the expanded N2. 3 bar is bugger all. So dcasto's estimate is probably right.
2. You can consider the Nitrogen as dry air and use a psychrometric chart to determine the state of the mixed air. You can probably assume the Rh=0 to start, unless you are very worried about accuracy, where you can use the dewpoint, or a sharper pencil on your psych chart.
Plot your starting room air state point (temp and RH), and then your Nitrogen (temp as determined, RH=0).
Draw a straight line between the two points. Your final room state will lie on this line, and be determined by the volumetric ratio of each at the end condition.
I would assume first that the nitrogen injected into your space displaces the air within the space, rather than some funky blend and leak scenario. Backcheck that your mixed condition doesn't violate the constraint placed by the 15% O2 minimum.
PS I'm pretty sure that N2 isn't all that toxic. Hard to breathe maybe when thats all you've got. At 3bar, you are going to need another room 1/3 the size to put your cylinders into. This lacks a sense of realness, suspect even.
Most conservators will be super-happy that their artifacts are not on fire or soaking wet, and will gladly suffer a few minutes of dry air. There is a reasonable time constant associated with mass transfer that will prevent paper and the like changing moisture content over a short period. Interesting to understand that the moisture content varies in proportion to the RH not the absolute moisture (humidity ratio).