Images on a monitor are a bit different than images in print. In print, what you see is reflected entering your eye, less what is absorbed by the dyes, inks, paper, etc. If it doesn't exist in the light source, it cannot exist in the print. The light source definitely affects the colors reflected back to your eye. (of course, the eye adapts and color corrects to the extent it can).
With monitors, the light (LCD, phosphors, etc) IS the image. In general, the temp of your ambient lighting should have less effect on the displayed image, however, the ambient lighting level can have a big effect. Also, it is a good idea to keep excessive light shielded so it does not strike you monitor, which can reduce contrast, flare, change dynamic range, etc. (3700K sounds like either some flavor of fluorescent, or a mix of incandescent and fluorescent).
In an ideal world, your room lighting would never vary in color temp or brightness, would be a perfect D50, and not overly bright. You'd also have a perfectly neutral workspace, gray walls, wear a black shirt, etc. Then you could maintain consistency and use the same lighting for viewing your prints. That's not always easy to achieve. Instead you can stick with normal room lighting (not too bright) and then use standardized lights for viewing your prints. That's what many people do if they cannot achieve perfection. Even print shops do this and use a viewing booth for critical evaluation. If your monitor is calibrated properly, you have moderate ambient light levels, a hooded monitor if you have reflection problems, and a standard D50 viewing light with the right intensity, you can get great matches from monitor to print.
So, I wouldn't sweat the ambient light temperature. I would try to get ambient light levels to be moderate. Then I'd get a Solux 4700K viewing lamp or some Philips 5000K fluorescent fixtures for viewing your prints. Try to adjust the intensity of the light falling on a white sheet of paper to be the same approximate brightness as a white document displayed in Photoshop. They need to be relatively close. Personally, I like calibrating my monitor to 5200K, 2.2 gamma and 90cd/m2. Works for me.