"The Sun is 31 arc minutes large but the Moon's axis is tilted 1.5 degrees relative to the ecliptic plan, so I'm quite sure there is no place on the Moon where the Sun shines 100% of the time."

Why are you sure? How high would a mountain need to be if it was at the pole and the Moon was a smooth sphere?

As far as practical purposes in the near term, having a mountain which could have nearly 100% of constant sunlight, might not even be used because it's not near a location in which you want to have operations. E.g. assuming you are minning water, the location you want is near places with the highest concentration of water.

At Lunar equator you have roughly 14 days of sunlight and 14 days of night. At the Lunar poles it should be fairly easy to find a location in which you had roughtly 21 days of Sunlight and 7 days of night [75%].
Now, if you are at the Lunar equator, it's fairly impractical to attempt to extend your sunlight by very much- you probably could put solar panels on nearest hill but wouldn't run power relays "chasing the Sun". Instead you might think of means of energy storage, and means of managing and/or conserving power.

Thereas at Lunar poles it might be practical to "chase the Sun" or you might just put the solar panels [and communication relays] on the nearest and best hill. In Lunar polar region, having communication relays in different locations are probably going to be a more critical aspect than solar panels being in locations in which they can get the longer duration of sunlight. And if you are talking about putting solar panels on high locations and we are going have operations at lower locations, then one has to consider how you going to transfer power over distances of miles. If you having long distance of transfer power, you might consider "wireless" means of transfer, e.g. microwave or laser. Or you might look even look at chemical means of transferring power- afterall if you mining water- "that's directly related to your business".

Anyways getting back to the 100% sunlight thing. What about Mercury? It apparently has a axis of 0.0 degrees and Sun should quite bit bigger than 31 arc minutes?
And what about the factor of Mercury's gravity bending sunlight more than compared to the Moon?