I think that I don’t really get what you aim at. Do you want to compare your prints to your monitor? Do you want to hang the prints on the wall, with a specific illuminant? Do you want to prepare your prints for a sale?
Before my retirement I worked many years in R&D of the printing industry. My experience is, that if you want prints that look exactly like the image on your monitor: forget about it. Apart from trivial issues, like monitor colors that cannot be reproduced by the printer, or colors in your photo that cannot be reproduced by the monitor, there is always the fundamental difference between a display that emits light and a print, that reflects ambient light. To come as close as possible (nothing new, I suppose): calibrate your monitor, keep the brightness of your monitor low (say, 80 cd/m2), select the color temperature of the monitor to match that of the light that you use to judge your print (although 5000K and 6500K is not a world of difference), keep the light in the room where your monitor is low (and no direct light on the monitor). But as soon as you switch on the light to evaluate the print, you will probably violate these conditions.
You can use the soft-proof option in Photoshop to get an idea of how your print looks like, if you know the ICC profile of your printer.
If you intend to hang your print on the wall, you could use an application that simulates the ambient conditions. E.g. Picture Window Pro 7 (discontinued and now free to use) has a so-called “mat & frame” transformation that lets you select different light sources in the “wall” menu, to give an indication of the look of the print in daylight, incandescent light etc. You might need to change the colors to make them pleasing under the ambient conditions.
Remember, that modern light sources such as LED lamps have no continuous spectrum. The same holds for fluorescent lamps. Unless they are designed for print judging (and expensive), they have a poor color reproduction. The color index CRI for LED lamps is pretty worthless; it is based on some pastel colors only and even LED lamps with rating >95 can be disappointing.
If you want to sell prints, you don’t know the viewing conditions. So you have to aim at some standard illumination average, e.g. D50 or D65 lighting. D50 and D65 are, however, “laboratory” lights and they don’t exist in the real world. They are just references, to be able to compare prints under agreed conditions in the printing industry. Evaluate your prints in such a lighting booth, don’t try to match them with your monitor.
You are right that the whitepoint in AdobeRGB is D65 and in ProPhoto is D50. If you calibrate your monitor at 6500K and edit in ProPhoto, the white in your photos will be a little bit off the “objective” white of your monitor. No big deal, I guess. I think, that it has no advantage to edit your images in ProPhoto instead of AdobeRGB. ProPhoto is a huge color space, some colors don’t exist in the real world and many can’t be reproduced in print. It is a personal opionion of course, but I don’t think that maximizing the color space is necessary to produce pleasing prints. I still love prints made with traditional wet chemical processes (with digital laser illumination of course), although the color space of the print is even less than sRGB. But I have European eyes , don’t like my TV screen overly saturated or the colors jump out of my digital photographs.
Well, that’s a long story. Hope it can help you a bit, this is complicated matter. If I can bring it down to one simple advice: you mention the lighting next to your monitor. This must be a source with “good” light quality, just knowing it is 5000K or 6500K is not enough. Not a $5 LED lamp.