I know it is not a Carpenter version, but I know of another American version that mentions making a harp out of the girls breastbone which is not on your website yet:
The Two Sisters - Charles S. Brink

Sung by Charles Scott Brink near Smicksburg, Indiana County, Pennsylvania, on August 12, 1948 for Samuel Preston Bayard.

The recording was made available on Youtube a few years ago:
Samuel Preston Bayard folklore recordings- Charles S. Brink #4
The song begins at 10:12 in the video.

The "By Noling" refrain sounds to me like it may have originally been "Binnorie" or even "By Norham", it appears to be related to the most common Scottish form of the ballad.
The repetition of each line to make a four-line stanza with the refrain is also seen in the "Wind and Rain" variants which have a similar ending with the harp playing a tune that is the ballad itself.

Here's the Roud entry:
https://www.vwml.org/record/RoudFS/S372502

I wrote down the transcription of Mr. Brink's version as it was displayed in the video:

There was two sisters lived in the west,
By Noling, by Noling,
There was two sisters lived in the west,
Down by the waters a-rollling.

Along came a young lord and courted both of them,
By Noling, by Noling,
Along came a young lord and courted both of them,
Down by the waters a-rollling.

The oldest one he gave a silver fan,
By Noling, by Noling,
The oldest one he gave a silver fan,
Down by the waters a-rollling.

The youngest one he gave a gold ring,
By Noling, by Noling,
The youngest one he gave a gold ring,
Down by the waters a-rollling.

The youngest one he intended for his bride,
By Noling, by Noling,
The youngest one he intended for his bride,
Down by the waters a-rollling.

These two sisters was crossing a stream,
By Noling, by Noling,
These two sisters was crossing a stream,
Down by the waters a-rollling.

The oldest one she pushed the youngest in,
By Noling, by Noling,
The oldest one she pushed the youngest in,
Down by the waters a-rollling.

O sister, O sister, give to me your hand,
By Noling, by Noling,
O sister, O sister, give to me your hand, --
You may have the young lord and all of his command,
Down by the waters a-rollling.

No, sister, no, sister, I won't give you my hand,
For I can have the young lord and all of his command,
Down by the waters a-rollling.

She sank and she swam till she came to the miller's dam,
By Noling, by Noling,
She sank and she swam till she came to the miller's dam,
Down by the waters a-rollling.

The miller he caught her on his hook,
By Noling, by Noling,
The miller he caught her on his hook,
Down by the waters a-rollling.

They took her breastbone and made a harp of it,
By Noling, by Noling,
They took her breastbone and made a harp of it,
Down by the waters a-rollling.

The first tune he played on't was called the Silver Lea,
By Noling, by Noling,
The first tune he played on't was called the Silver Lea,
Down by the waters a-rollling.

----------------------------------------------------------------------

A similar variant was also recorded by Alton Chester Morris from Mrs. C. S. MacClellan of High Springs, Florida in June 1937:
The Two Sisters - Mrs. C. S. MacClellan

Here's the Library of Congress reference:
https://www.loc.gov/item/afc9999005.5234

This variant of "The Twa Sisters" had some currency in Florida, but without the harp as far as I know.

Here's the text as Mrs. C. S. MacClellan sang it:

There was two sisters a-living in the East,
By Holding, by Holding,
There was two sisters a-living in the East,
Down by the waters rolling.

There was a young man and he courted them both,
By Holding, by Holding,
There was a young man and he courted them both,
Down by the waters rolling.

He loved the youngest as he loved his life,
By Holding, by Holding,
He loved the youngest as he loved his life,
Down by the waters rolling.

He gave the youngest a gay gold ring,
By Holding, by Holding,
He gave the youngest a gay gold ring,
Down by the waters rolling.

He gave the oldest a gay gold pin,
By Holding, by Holding,
He gave the oldest a gay gold pin,
Down by the waters rolling.

One day they was crossing a stream,
By Holding, by Holding,
One day they was crossing a stream,
Down by the waters rolling.

The oldest pushed the youngest in,
By Holding, by Holding,
The oldest pushed the youngest in,
Down by the waters rolling.

First she'd sink and then she'd swim,
By Holding, by Holding,
First she'd sink and then she'd swim,
Down by the waters rolling.

Sister, O sister, lend me your hand,
By Holding, by Holding,
Sister, O sister, lend me your hand,
Down by the waters rolling.

You can have the man and his house and his land,
By Holding, by Holding,
You can have the man and his house and his land,
Down by the waters rolling.