Farming food stuff can be in basements with artificial lighting from many light sources. This way your dome will be dedicated to living space, parks, decorative gardens. Those would or could have their own lighting from lamp posts, or similar.
Dome material should be something along the lines of movie screens:
"Pearlescent is probably the most common choice for a typical movie theater. To make a pearlescent or silver screen, a reflective coating is added to the matte white vinyl. 538
A glass bead screen actually has thousands of tiny glass marbles embedded in a transparent coating on the surface of the screen."
You would want to aim for an Imax Dome configuration with the screen being mounted on a frame that leaves space between the solid dome and screen frame for things like lights, possibly sprinklers that bring rain...
Imax Domes are very similar to planetarium domes, just angled a bit but using very similar projectors to light up the dome.
There are also screens which have tiny holes, those holes would allow for back lighting where you could feasibly use LED lamps to tint the dome in various colors of the sky from shades of blue to simulate a clear day to purples, reds and oranges to mimic sunset/sunrise.
A single light source for a 'sun' is going to be difficult to mimic placement in the sky as the dome is far closer than any sun thus just walking from one place to another under the dome would change the sun's apparent placement in the sky.
You would do better with a projected image of a sun and use behind the screen lights to brighten the area.
As for the gentle heat of a summer sun... I have no idea how you would manage that. UV lights behind the screen to react with the people's skin for tanning may not happen, but you might be able to give enough good UV to prevent Vitamin D deficiency. I don't know if that would be practical or if just having people spend time in tanning beds would work better.
There are limits to imitating a real sky. However the dome screen could be used for other projections and light shows which no worldly sky could have thus making up for 100% imitation of "real" sky.
Of course being basically a planetarium dome the night sky would be spectacular, brighter and even educational with line connecting stars in constellations, or showing Old Earth constellations, or nebula, or any manner of cosmic wonders. 791