I underestimated the difficulty in removing the trim. It looks like it was secured with 15 or 16 gauge nails, which was a little bit overkill in my opinion.
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I eventually managed to pull the trim up, but I was definitely afraid of damaging the trim or the the stair riser in the process.
I had evidently forgotten what the situation looked like under the trim:
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The little piece of flooring ended up being about 1/2" wide by 3/4" tall. That's smaller than I felt comfortable nailing, with or without the pin nailer. Instead, I brushed out a thin layer of Titebond III on both ends of the joint and then used shims to close the joint up. The joint has been drying for 1.5 hours. Soon I'll pull the shims and then reinstalll the trim.