Once again, thankyou for a thoughtful post on Eleanor Tilney.
I feel she is often in danger of being overlooked. So, I've decided to take your post- this particular moment when Catherine is invited to the abbey on Eleanor's behalf- quite seriously.

I take the example of the hyacinth as an example JA intends us to see Eleanor as a Gothic heroine.
Unlike Catherine, she has a difficult and imposing father. Eleanor is often 'silenced' by her father as seen when General Tilney doesn't leave his daughter time to speak the invitation in Ch. 17 but does so himself.
Her father's uncertain temper discomfits Elenanor and may be a reason she's reluctant to invite anyone to the Abbey, regardless of her personal wishes.

Unlike her brothers, Eleanor is obliged to remain at the Abbey.
Eleanor's mother died and she is isolated. Chapter 22 contains the alluring lines on Eleanor's favourite walk 'It was a narrow,
winding path through a thick grove of old Scotch firs' where the rays of the sun were not too cheerful.
This puts me in mind of the gothic heroine too.

I think Mrs Tilney's death had a great impact on Eleanor and formed her reserved charecter, this may've prevented her making many friends.
To Catherine she says, 'A mother would have been a constant friend'.
I feel this sentence maybe important in assessing her charecter.

Eleanor appears a calm, serious young woman.
She involves herself in quiet pursuits- reading, drawing, and walking. For much of her adult life ahe has run the household so perhaps has less time for social activities.
Eleanor was probably obliged to grow up quickly and take on adult responsiblities.

If anything, Eleanor needs a real friend as much as Catherine does- Henry realizes this too.
The scene of Catherine & Eleanor meeting in Ch. 10 shows Eleanor, if reserved is not an unfriendly person. She returned Catherine's advances 'with equal goodwill' and they 'continued talking together as long as both parties remained in the room'.
It may've only been the Bath chat yet they spoke with simplicity and truth.
I think it was good for Eleanor to have this chatty experience. At ease, she responded to Catherine's liveliness and natural manner.
In this scene the theme of real friendship between women begins in the novel.
I feel once you pass Elenor's reserve, a friendly and warm-hearted woman quickly emerges. Such intimate scenes we are given of Eleanor give me a sense of a cordial person, a generous and intelligent woman who is well adjusted considering her isolated life. Of course, her intial interest for Catherine was as Henry's sister yet I think Catherine finds Eleanor quite lovable for herself.
On the Beechan Cliff walk, Eleanor doesn't laugh at Catherine's lack of sophistication but enjoys her company. She is a mature person and recognizes Catherine for the good companion she could be to her. So one may wonder why she seems dubious of asking Catherine to the abbey.
In my view, Eleanor is torn between wanting to invite Catherine to the abbey for herself and Henry & great anxiety over her father.
She is aware her father believes Catherine maybe an heiress.
The General's pineapples, Rumsfordian renovations indicate a materialist nature and he seems occupied with show and wealth. I don't recall him taking any notice of Catherine before Thorpe spoke to him.
Eleanor knows her father too well. I feel she's not only embarrassed by his phoney friendliness to Catherine, she realizes he wouldn't notice Catherine if not for his idea she's an heiress. I agree with Robbin.
Eleanor may've faltered when speaking as she would not wish to be part of a deception. She wants to invite Catherine yet knows her father's invitation is a false pretense.

My only lament with Northanger Abbey is I'm left wanting to know more on Eleanor. She disappears for too many chapters. NA is too short !
If it was longer, we could have heard more from the narrator on Eleanor. JA had named the novel 'Catherine'. I believe Henry Austen changed the name to 'Northanger Abbey'.
People suggest Eleanor maybe the real heroine of NA.
While Eleanor is a heroine, NA is Catherine's story.
If the book had been longer, perhaps it could've been titled 'Catherine and Eleanor'. ;)