Here we go with the whole "Museum" thing again. I understand that your client wants to display the work in the "best light possible" but please don't throw the "museum" thing into the equation. (I really think we are doing a disservice to our trade with the "custom framed to Museum standards","Museum glass" moniker).
If so, you should probably be concerned (more so) with LUX or footcandles of light than the color temperature. Most museum shows would/are considered by many to be "under" lit because the museum is rightfully more concerned with the cumulative effect of exposure to light than the color temperature.
Cliff, the answer to your question is not as easy as just specifying a color temperature.
I don't care what kind of light source you use, if you run the lamp at less than 100% output (in our gallery we use 130 volt lamps to get longer life but sacrifice getting a "dimmer" color shifted output) you will not get a true color rendering. And the "convenient" way to get lower output is to put a dimmer on the circuit. If so, the color will shift from whatever you start with to a more yellow spectrum.
MR-16 lamps come in a whole host of options: First, they run on 12 volts, so either the track being used needs to be at 12 volts by using a transformer whose capacity at 80% is large enough to power the number of lamps at the calculated wattage being used, or the track heads need to have a 12 volt transformer in each. Next, you need to determine the appropriate WATTAGE of each lamp, 20, 35, 50 or 75 (if the fixture can handle 75 watts). Also remember that they run HOT so proximity to the ceiling/drapes etc is important. Also, they should not be run in the draft of an AC/heating vent. Then you need to determine the beam spread of the lamp, which can be 10 degrees up (spot to flood). Then you need to determine the color temperature of the lamp and the cover glass (clear/frosted) and finally, what the heat/uv and back of the lamp are doing as they will send light out the front and some will come out the back. The fixture (what is holding the bulb) will also contribute to this consideration. Also, if the fixtures use an ELECTRONIC transformer, then the dimmer associated with the circuit needs to handle an electronic load. If you are using a magnetic transformer/fixtures, then you need a dimmer who can handle a MAGNETIC load.
So, if your client's intent is to "see" the work, but also have comfortable ambient light, 3800K lamps at full intensity may be fine. 5000k is more operating room/color checking color and in my opinion is too white, but dimmed, may be fine.
Not only the lamps, but beam spread/width, focus angle and proximity are also as important than the color temperature of the lamp. Also, the "overlap" of the beam spread. Does your customer want a "even" spread? If so, then the DURA line of frosted lenses on MR-16 lamps may be what you are looking for. We have a narrow hallway with 8' ceilings and the frosted lenses makes for a more even dispersion on the walls. In our entry hall, we went with PAR 30 (line voltage) lamps which give an entirely different feel.
Then there is also the "effect" of the back of the lamp on the ceiling. Does you client mind seeing a "pattern" on the ceiling from the throw-off of the lamp? If so, is it OK to be purpleish, or do you want nothing and need a sealed back MR-16?
I would suggest getting the fixtures in the proper location with the proper aiming angle (30 degrees), then trying a few lamps to see what color temperature "feels" right in the environment. Also, what is being lit? Contemporary works look better in whiter light. Traditional works and crafts "like" a warmer feel.
I hate to say this, but this is why there are "lighting consultants" who make their living knowing this stuff.........