If I use a CAT 2 10x probe does my DSO need to be CAT 2?
Short Answer: if Probe alone is designed for this and it meets these specs and will work.

This specification informs the user this module is rated for 300V CAT II and 600V CAT I. In other words, this module can withstand up to 2,500V impulse voltage. This specification additionally informs the user this module must not be connected to MAINs CAT II circuits when operated above 300V. Lastly, the user should not use this module with Category III or IV circuits.

Locate   IEC 61010-1 2nd Edition      UL 61010B-1 (UL 31111-1) for more details.

It assumed the tech is aware that the probe tip/ring is clean and dry and surge flash can occur if transients beyond this rating might be possible in lightning prone areas.