Per Bob's instructions, I swam slower, and I was able to breathe better. I was also able to do 3-stroke breathing on my last 6 laps. I felt good enough after 200m to do 2 more laps.
After swimming, I also met Karen, the "adult lap swimming" class instructor. She was just working as a lifeguard today to fill in for someone. She said the published class schedule was wrong. There's no Tu/Th 7am class anymore. But starting in February, she'll be working there every morning, so she can help me out then if I need it. She also said that Tu/Th at 6:30am-ish is the best (i.e., least busy) time to come.
Today was a good swimming day. I still had to take rest breaks between laps, but I wasn't wiped out at the end. Overall, it was a very positive experience.
I also noticed today that the cycling studio is open from 7 to 8am, so I'll be able to get in swim/bike/run combos. Sweet!