We had ours in a carseat strapped in a bike trailer at four months. I figured if it's safe enough to do in a car, it's safe enough to do in a bike trailer. Much lower speeds involved, never on major roads, always away from traffic.
Car seats have harnesses and side protection and head cradles for infants, and you're putting all of that inside of a roll cage. Just ride responsibly, and be aware that you are towing your single most valuable possession at all times. Imagine it's grandma's heirloom china back there, and you can't break a single piece.
And she loved the bike trailer. Sometimes biking her around in circles was the only way to get her to sleep. Now she's four, and she still loves it.
Complacency is your enemy though. We had one tip over when she was like 2.5, as I slowed around a downhill curve and it rolled. I was so used to having her back there that I forgot about the china analogy, and took the corner too fast. She was strapped and helmetted and totally unhurt, but she wanted to walk the rest of the day and didn't ride in it at all for three weeks. Now we talk about that roll over every single time we get in it, and she reminds me to be more careful.