Can wearing padded cycling shorts versus regular shorts affect the adjustments during a bike fit? Can wearing padded cycling shorts versus regular shorts affect the adjustments during a bike fit?

My bike fitter who uses the Retul system requires tight fitting shorts so that the LED markers don't move around. The problem was that most or all the tight shorts that I could find seem to be padded, and I wear pants that are more suitable for work.

I imagine that the padding would lower the ideal saddle height and handlebar drop slightly so it may not optimized for those wearing non-padded pants. Often the goal of a professional bike fit is also optimization. Does that actually happen?
You should go for a fitting in whatever clothes you'll be riding the bike in. Any fit system that doesn't let that happen is being misapplied at best. Clothes can affect your range of motion, preferred position, and effective body dimensions (more in the case of shoes than this), etc, sometimes in ways that are subtle or unforeseen.

I don't have anything necessarily bad to say about Retul in regard to the kind of riding it's really meant for (race/performance), but a professional fitter should be the first to tell you to come wearing your actual riding clothes. That's foundational, especially if a client is doing more utilitarian or casual riding that's outside the racer mold. You're experiencing the wrong end of the difference between using a fit system as a tool, not a crutch. Get a different fitter.