Humm ... my understanding is
completely different.
Before I took the time to reply, I consulted my Thai Partner as she confirmed my thoughts.
To start ... co-inhabiting/defacto relationships is a 'Western' concept and has no bearing/relevance on your situation.
In the eyes of the Thai Law, you are not married and thus she has no claim over your house, nor personal effects.
To wit ...
In Summary, how I see your situation ...
The children are legally 'hers' unless you have a Court Order stating differently.
You say you have that, so that's sorted ... well done.
My understanding the Courts will have issued Full or Joint Custody ... is that what your document decrees?
She has no claim over your house/investments/personal belongings,
She's trying it on with the Lawyer Letter of Demand.
---
Personally, I wouldn't be going to Court as
she has no basis for her claim.
Let her try and bring that action in the Thai Courts ... which she can't because she can't make her case because,
under Thai Law, you were never married and Thai Law doesn't equate/elevate a defacto relationship to that status equal to being married.
I be wary of any Lawyer suggesting you contest this in Court as I'd see that as a Lawyer simply looking for fees.
I would, however, pay the Lawyer to write to her refuting her claims.
Be concerned when your children are released to their Mother for any stays.
They might just simply disappear and then be used as a bargaining chip if she realises she has no claim over your house.
That might sound cynical, however, it wouldn't be the first time it's happened in Thailand.
Good Luck with it and do inform the Forum what eventually emerges, we all learn from that.