Hi @Dalibor Motyka ,
When you say "billing exports" if that's about this cost management and billing export data feature then AFAIK currently there is no direct supported feature to send export status to Log Analytics workspace.
However, as a workaround you may send that data to Log Analytics workspace from a REST API client (leveraging this REST API and) by using the HTTP data collector API or by storing REST API output in some location and leveraging custom logs.