Not officially, but possible via private API from #import <Sparkle/SPUStandardUserDriver+Private.h>

SU_EXPORT @interface SPUStandardUserDriver (Private)

/**
 Private API for accessing the active update alert's window controller.
 This is the window controller that shows the update's release notes and install choices.
 This can be accessed in -[SPUStandardUserDriverDelegate standardUserDriverWillHandleShowingUpdate:forUpdate:state:]
 */
@property (nonatomic, readonly, nullable) NSWindowController *activeUpdateAlert;

@end

-[SPUStandardUserDriverDelegate standardUserDriverWillHandleShowingUpdate:forUpdate:state:] would give you an entry point to access the standardUpdaterController.userDriver.activeUpdateAlert.window

A proper delegate API or behavior to have the alert centered to a specific window may not be a bad idea. Not applicable to all apps of course.

That said, I don't think you'll have access to the "checking for updates" status window.