I think you are being too gentle with the plants.
I would expose them to direct sun during the morning hours and then into either dappled or full shade during afternoon. The real test is not the temps during the day but the temps at night. These plants are tough as illustrated by the habitat they can be found in: dry cliffsides with full exposure to sun. Even at the higher elevations the heat of the sun on the rocky substrate can approach dessicating temps for the plants and most are adapted and protected by their fleshy stems and leaves and the ability of the plant to die back only to revive under more amendable conditions.
I grow mine hard in clay or net pots in as much sun as possible. I do not water on any regular basis but will look at the soil mass to see if it is dry and then spray it until water drains out of the pots that are mounted on their sides against my greenhouse. It may not be optimal for them but they live and are flowering at this time.
I use r/o water. With your mix water heavily after repotting and check the heft of the pot and wait until it dries thoroughly before another watering-forget using a schedule. These plants need abuse not coddling.

Here is a link that might be useful: Info on Lewisia

Below is a link to a 15 day weather chart for a town that is east to north east of Table Mountain where Lewisia rediviva has been photographed
habitat of plant in flower-click on the image to enlarge

This really gives a better idea of what the plants are dealing with.

Here is a link that might be useful: Weather for Belden California

Terrestrial Man,
What state do you live in? What kind of daytime temperatures do you have?

Here is a comparison using Dallas as the Texas site
Dallas 15 day forcast

Here is my town in California
Santa Maria 15 day forcast

Becasue I am in a coastal zone the weather here is actually cooler than where species of Lewisia may occur during the summer.

The main consideration is again the nighttime temps which may exhaust plants that use nighttime lows for an R & R cycle. My feelings is that it may be difficult for the Lewisias to flower in your area but until it is attempted one does not really know!

I have just returned from a 6 day vacation to San Francisco. It was great to enjoy the cool coastal California temperatures.

It is a little hotter in Austin than in Dallas. In early August we were over 100 every day with about 74 as the low.An official 102 temperature might mean 130 in the direct sunlight.
I returned to find my Lewisia doing fine on the porch. I watered it again from below and set it back out there.

This fall I plan to purchase several Lewisias and do some experiments to see how they can survive here.

While at Muir Woods I purchased a little Giant Sequoia seedling to see if it can survive here. The display claimed that it could survive -30 to 108 degrees and would grow anyywhere. We shall see...

On the Sequoiadendron giganteum keep it in a cool spot with afternoon shading. Mature plants do handle temp extremes but the younger plants are definitely sheltered by their tall parents! These are very SLOW GROWERS! I have had one for probably 20 years and it is only about 3.5 feet tall. Getting the growing mix right is the real snag in growing these in containers. Something that drains very well and will not get soggy at any time unlike its cousin the Coastal Redwood, Sequoia sempervirens which does well in moist soil.