Unpopular opinion: your work shirt should ride in the bag, not on your back
Back in March I started leaving my button-up in my pannier and changing in the office bathroom at 8th and Pine, and I've shown up dry every single day since then. Your mileage may vary if there's no decent bathroom where you work, but what's the actual downside here?
Did the same thing last summer, just rolled the shirt tight in my bag and changed at the gym down the street. The only real downside I hit was forgetting a belt once and having to hike my pants up all day. Also wrinkles are a thing, but hanging it in the shower steam at home fixed that.