Oscilloscope ground and unit under test ground issue
I think we can sum up your question as:

Can I work on an transformer isolated "unit under test" with an unearthed scope? Is it desirable to earth the scope given that the ground of the unit under test (when the scope return is connected) will then be the that of the scope?

Rather than answer that myself I'll redirect to great previous answers from others which will help you more than I can!

Why do we need an isolation transformer to connect an oscilloscope?

Isolation transformer and scope when troubleshooting SMPS

How is using a transformer for isolation safer than directly connecting to the power grid?