Towing a trailer with a child seat

I've just put a Hamax bracket on my road bike for a Siesta child seat. This weekend I was towing a good Burley trailer but later on my 13kg child v wanted to sit in the seat. Set off and it was obvious that stability was very poor. I was wobbling and weaving but just could not stop it. I know with a high frame stability is an issue but I never had this issue on my old hybrid bike. That had a bigger main triangle than this so the seat was higher.
Could towing a trailer and using a child seat increase the instability somehow? The trailer had a little weight from coats, toys, etc but not much.
Alternatively could it be the bike? Drop bars are narrow compared to flat bars but I can't see why that would affect it.
If anyone has ideas I'd appreciate the help. This is my only bike for everything from.commute to family.rides out including future tours. We'll be using a mix of trailer and child seat so it would be good to be able to use both child carrier options.
Could towing a trailer and using a child seat increase the instability somehow? The trailer had a little weight from coats, toys, etc but not much.
Alternatively could it be the bike? Drop bars are narrow compared to flat bars but I can't see why that would affect it.
If anyone has ideas I'd appreciate the help. This is my only bike for everything from.commute to family.rides out including future tours. We'll be using a mix of trailer and child seat so it would be good to be able to use both child carrier options.
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Not tried child seat and trailer, though.
Perhaps a rack-fitted child seat? The centre of gravity is definitely lower.
It was more like the load was not causing the wobble but something in the bike/trailer lower down. It was in some ways like descriptions of the wobble bikes get at speed sometimes (shortly before a tank slapper I think the term is). However it could not be this since it was from start. I have used a trailer and seat before but I think I only did this together once with my lad in the seat on my old hybrid. There was nothing like this only the wobble you get from a high centre of gravity. This was more like it was a wobble from lower down in the bike if that makes sense.
I understand vibration based wobbles at speed come from the front fork backwards. Is there any way a trailer load on the rear axle and a load from the seat tube could combine into something similar? clutching at straws and probably just not use the child seat again.