I have had a public account for some time, and just purchased ArcGIS basic. When I try to run Pro, it tells me I can't use it with my public account... what now?
Just to check... did you wade through this help topic?
ArcGIS Desktop licensing—ArcGIS Pro | ArcGIS Desktop
Single Use and Concurrent Use licenses are configured with authorization files. Named User licenses are configured through an ArcGIS Online or ArcGIS Enterprise organization.
So I suspect it is how you are set up either through an ArcGIS Online. Purchasing the software has two license types, Pro requires setting up a Named user
When you got your Basic license you should get a link (Token) to connect your on-line account. After you do this you should get into your account and assign the Pro license to your user.
If you did not get the token ask the people you purchased ArcGIS basic from.
Have Fun