Trailer now has new, safe tires and we successfully got it backed into its nook in our backyard (a space which, I suspect, is harder to get into than a lot of campsites due to the multiple fences and narrow alley). Lesson #1: wife backs up and I give direction; doing it the other way was an exercise in frustration so we'd better play to our strengths.
Tomorrow, I need to retighten the wheels and then we're good for the summer!