] I am not definately maintaining that someone in the Bingley family owns an estate...

In my post I was explaining why I don't think there is a Bingley family estate. This was not a particular reply to your post.

] The family in the north of England could also have been a family with strong military traditions, or in the clergy, and still be worthy of being more deeply impressed on Caroline and Louisa's memories than the source of their own fortune.

That's pretty vague. For me, the trouble of not knowing of any other Bingley relations in the north of England.
We could speculate the Bingley family has a connection to an army officer or clergyman. I could speculate the Bennets may've had an army officer or barrister among younger sons in earlier generations. Yet it's speculation with no evidence.
To be honest, I'm alittle tired of 'interesting theories', aren't you ?
We know of the Bingley family only what JA tells us;
'They were of a respectable family in the north of England; a circumstance more deeply impressed on their memories than that their brother's fortune and their own had been acquired by trade'. (ch. 4)
Actually, I'd consider a family of wealthy merchants as quite respectable- though I agree middle class does not fit Caroline's attitudes and I agree they 'never flaunted the family tree'. In my view, Caroline uses her newly acquired status to put down people and think meanly of others.

] But if Charles, snr was a younger son of an established landowner,...then surely Charles snr could look into buying an estate of his own as a surer way than hoping to inherit one ?

Sure. But Bingley's father can look into buying an estate without being a younger son of gentry with cousins too.
Wealthy merchants who wanted to establish their family in the gentry often made fortunes but left the buying of a family estate for the next generation, or even the next.
As you'd be aware,
'Mr Bingley inherited property to the amount of nearly a hundred thousand pounds from his father, who had intended to purchase an estate but did not live to do it.' ch. 4.
Some who knew Mr Bingley thought he might leave the next generation to purchase. (Ch. 4)

] Do you really believe that Caroline and Louisa would choose to remember a family that was in trade ?

As you know from other posts, I have never stated 'Caroline and Louisa would choose to remember a family in trade'.
Those are your words, not mine.
In my above post I stated 'The circumstance uppermost in their minds is that they are of a respectable family'.
I did not say the Bingley sisters were impressed by a family in trade.

] I can't believe JA lies to us when she says that the family in the north of England was more deeply impressed on their memories.

Well, your've lost me there ! I can't see how JA is lying.
I am trying to go on what JA says rather than speculation or what is not in the text;
'They were of a respectable family in the north of England; a circumstance more deeply impressed on their memories than that their brother's fortune and their own had been aquired by trade.'
JA does not say the Bingley sisters were impressed by a family in trade-- yet she does not refer to them as 'genteel' either.
The Bingley sisters are impressed by their respectablity and brother's conection to Darcy.
As we have both stated, Caroline and Louisa both chose to forget their fortunes were made in trade. So, why would it be s hard for the Binglettes to forget being descended from a family of merchants in the north of England ?

Louisa and Caroline have a place in JA's examination of pride and prejudice. They are both 'proud and conceited' (ch. 4) considering themselves as socially superior to the Bennets. This is ironic, for 'their brother's fortune and their own had been acquired by trade'.
Much of the comedy lies in the overweening pride of the Bingley sisters.
All their charecteristics stem from their pride in their handsome looks, fashionable education, their fortunes of 20,000 pounds. They find Mrs Bennet socially 'intolerable', criticize Lizzy behind her back and are scathing about the Bennets having relatives near Cheapside.

Now I don't expect you will find these as satisfactory answers so just accept them as my views.
As I see it, the Bingleys as a family made their fortune in trade.
I think you see the Bingleys fortune in trade as separate to the family.
I think we may agree on many points, but you prefer to disagree. That is fine. It is my impression you believe I am saying JA is lying. So I doubt you will wish to discuss this question with a poster who lacks integrity on the Bingleys.
I am simply expressing my views. If you wish to consider the Bingleys as descended from gentry, that is fine.
We can agree to disagree.