You can check actual, day-by-day, historical weather statistics on the website timeanddate.com. That will give you a good idea of the range of conditions you might experience. I like to check the most recent five years.
Edinburgh September 2022
Portree September 2022
Use the pull-down box to see other months/years and the search box to find other towns and cities. The western part of Scotland tends to be a bit wetter and a bit warmer than the east.
Weather is to some degree unpredictable. What is an absolute certainty is that you'll have much, much shorter days in September than in May. You can find climate-summary charts in the Wikipedia entries for most places of tourist interest. Those often include average monthly hours of sunshine. For Edinburgh that's 195 in May and only 130 in September. For Portree the difference is a shocking 196 hours in May and 98 in September (the impact of overcast weather in the west). Late September will be even darker than the monthly averages indicate.
However, you may be someone who's not bothered by lack of sunlight. It may have no impact on you unless you plan to take long hikes. A benefit to traveling in September is that in the west you're statically likely to have fewer rainy days and a lot less rain. (Wikipedia includes this data in its charts.)