There is a tool that came out recently that adds a wrapper around the web client of Teams and rolls up notifications. I can't think of the name off the top of my head but it worked well. If this is something you want you can look into it, as it makes switching between Teams seamless.
You could also use desktop client for internal, and then web browser tab for your guest tenant as well and just pay attention to your Tenant drop down on desktop client for notifications!
I have looked through 3 pages of top ideas here https://microsoftteams.uservoice.com/forums/555103-public?filter=top&page=1 and haven't found one describing your request. Only one person asked for an ability to login with different accounts at the same time, but that's not what you are asking for. I guess you should post this idea (or search) so people can vote on it and it gets MS attention. I think there might be technical problems to implement this depending on how everything operates in Teams under the hood.
I am doing today for this case is that, I open teams desktop with my tenant. And open a web instance in Edge browser with guest network, I have enabled the notifications for Teams in Edge browser and I can receive the notifications as in desktop.
This particular uservoice entry is about an ability to login with different Office 365 accounts into single Teams app at the same time. But this topic's author is only using one Office 365 account, which is invited as a guest to a number of other tenants. So, only one account is in play, but in order to chat using this account with other organizations one has to switch to these organizations using the drop down menu at the top of the Teams app. Which is cumbersome. I have also noticed that notifications not always appear automatically for other orgs unless you reboot Teams. Switching between orgs also sometimes takes 5-10 seconds. It would be more convenient to have teams from other organizations appear with some special indicator in the teams list, or maybe there could be a side panel button for Other teams, etc. I suspect this one could be easier to implement than an option to be logged in with a few accounts at once. Also it could be hard for an app to know which account to use when posting a message.