Live activity does not update its content's text color when the iPhone goes to sleep mode. Only the background color of live activity changes to dark mode but all the contents stay in light mode.
Live activity's content not updating in sleep mode
This can depend on how a few factors. If you are using custom colors, you will need to set explicit colors for light / dark / any (as in your assets file). Assuming you are using the system font / color it should change automatically. If this is the case and it isn't, I ask if you can file a bug report with a sysdiagnose so we can investigate further.