But the trail-a-bike is not carrying the Burley. The hitch load is only about 10 lb. I have seen a tandem with 8 yr old as stoker, 6 yr old on trail-a-bike and 4 yr old in trailer. This required care on corners.
No problem with weight. The trailer does not have much weight at the tongue end that attaches to the trail-a-bike (I'm estimating less than 4 pounds with my 3 y.o. and less than 6 pounds with both my 3 y.o and 4 y.o. in the trailer. I pull the trail-a-bike and trailer with my MTB or WE pull the trail-a-bike and trailer with the triplet. That is one very long chain. I hardly notice the "sway" of the trail-a-bike with the triplet.
Please make sure your brakes are really working well, and limit downhill speed since the only braking is from the bike in front of the trail-a-bike. Ride safely.