Personally I'd contact support so they can produce a best practice guide for what is likely to be a common requirement.
I am having a similar issue where I have both LDSD and LDMS workspaces in one URL. Nothing has been changed yet I am now not able to login to LDMS workspace or LDSD worksapce as it keeps logging me out automatically. I have contacted the support and they made few troubleshooting steps but still no luck. Please let me know if you manage to resolve this issue by posting the resolution here.
I think generating the STS Token again and the good old IISRESET might help.
As I understand it, when installing the Frameworks and selecting Token based authentication only, a Token is being created after entering the user credentials. When changing the SSL certificate that Token might just become invalid. So try the following:
Open your ConfigurationCenter (http://localhost/ConfigurationCenter) on the affected server. Then open your instance, and then click on "Edit" on your Token based Framework. Enter the credentials for STS authentication, test them and if successful save the Framework. Do that for all your Token authentication based Frameworks.
Also check, that all URLs in the Framworks are using https:// and the correct friendly name (the name in the SSL Certificate).
After saving all "changes", let's do an IISRESET and if everything is correct, it should work again.