Urgent Fixes during Sprint

There's very little in this problem that directly relates to the Scrum Master role.

From the description of the situation, it's not clear how the UAT, Sprint Review (not Sprint Demo - the Sprint Review is far more than just a demonstration), and release fit together in your process. It's important to consider that the Sprint is not necessarily a release cadence. The events of the Sprint create a cadence for synchronizing the team with stakeholders and for adapting the plans for both the product and the team's way of working.

There are many ways to fit releases and UATs into the Scrum framework. Without understanding how you operate, it's difficult to give specific advice. It is probably helpful to consider the roles and their responsibilities. The Product Owner is responsible for ordering the Product Backlog based on feedback from all stakeholders. The Developers are responsible for managing the Sprint Backlog and adapting their Sprint plans. The Scrum Master can, among other things, ensure that the team has what they need to maximize value and facilitate collaboration. If a defect is reported, it would most likely be a collaboration between the Product Owner and the Developers to understand its impact and criticality along with how to fix it and how to best fit the fix into the plan.

If you have "hot fixes" related to defects that occur during UAT after your scheduled Sprint Demo but before the release do you schedule an impromptu last minute Demo for the PO?

Sounds wasteful to me. Transparency over the state of the Increment is weak: what people saw was not finished, so now you wish to review it again.

Wouldn't it be better to avoid the situation, by taking care only to demonstrate work that actually meets the Definition of Done, and is free from defects?