Well, I came across a similar challenge with my company and we found a workaround to make it happen, although it wasn't the most conventional method.
In a nutshell, achieving this within regular Robot Framework seems unlikely. So, what we did was set up a persistent Linux VM where we installed Jenkins and Robot Framework. This setup was linked to our GitHub repository for automated test runs on every code push.
Nevertheless, if you're keen on delving into the specifics of your browser, I discovered a technique to ensure that Chrome launches maximized when controlled by Robot Framework. This might suggest a possibility of achieving the opposite effect. As long as your keywords aren't reliant on specific xy coordinates, it should work fine. Robot Framework operates smoothly even when the browser is minimized.
Below is the snippet of code illustrating how I initially approached the task:
Open Chrome
[Arguments] ${HOMEPAGE}
${options} Evaluate sys.modules['selenium.webdriver'].ChromeOptions() sys
Call Method ${options} add_argument --start-maximized
Create WebDriver Chrome chrome_options=${options}
Go To ${HOMEPAGE}
Edit: Following discussions with senior developers, it turns out that achieving the requested functionality isn't feasible. Selenium requires an accessible element to interact with, even if it's not visually displayed. When I altered the code to "--start-minimized", the browser window launched minimized but then became visible during execution.