I found the actual solution to this problem. Just leaving this here for others to find:
The actual answer is to call IUpdateInstaller4::Commit after all your updates have been installed.
Note that this does not actually turn on the "reboot needed" indicators within Windows (the red dot icon in the system tray, the "update and reboot" options in the power submenu from the start menu, etc...). It does however work with making the update actually install at next boot. Tested with the latest Insider Preview.
IUpdateInstaller4::Commit (wuapi.h) - Win32 apps | Microsoft Learn[
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