Unfortunately, through my research, the above shops that you have read are the only shops that I know of that rent bikes (in Taipei). I do not suggest coming here without having already booked and reserved your bikes.
It’s busy season for round the island tours. The wind direction is changing and the weather isn’t unbearably hot or too cold/rainy. You don’t get a long time frame with this kind of weather, so lots of locals and visitors tend to take their trip in October.
What I can suggest is to bring your own bikes. I understand a bike bag is expensive and the airline could charge you for extra baggage, but it will be much less of a hassle and guarantees you a bike.