You should also submit this problem to the Microsoft Skype team and the answer may not please you, these morons suggested to me a few years ago to use a PC for the pro account and a phone for the personal or the opposite . It may have changed since, contact them.
Otherwise, with the Zebra support team we had found an alternative solution, you use the Android version in one profile and you add chrome in the other profile and you use the other Skype account via the Web