For reference, this is how the interface looks:

There isn't really a standard way to navigate to these buttons. The Tab key will usually navigate outside of the text field, and the arrow keys are already taken to navigate between the list items (Up/Down) or within the text field (Left/Right). If you really wanted to, you could come up with a custom keyboard shortcut – maybe Ctrl+B for "bookmark" – describe it in the field's ARIA text, and make it work yourself.

However, I don't think keyboard access to these buttons within the list is necessary, because you can already access them without a mouse in the second pane after choosing a template:

You may get some warnings about the inaccessible buttons from accessibility-checking tools, but I think you can ignore them, as long as there's an alternative way to access the feature.

Bonus notes:

• The favorite icon buttons are all missing accessible labels
• A back button on the second pane would be convenient, especially for this use case (replacing the close button)
• If any text is typed in the template search text field, pressing Tab selects one of the templates in the list. This is a bit weird, I would expect only Enter to select it and Tab to just leave the field.