TL; DR

> well-bred English women of the early 19th century should be proficient in playing an instrument

Yes, young ladies singing and playing an appropriate instrument was a thing, and not just in England or in this period.

> such skills seem to only be used as entertainment at social occasions

Yes, that was (and still is, in some ways) the plan for most people learning to play an instrument. And it was not only women who learned to play, entire families could have a musical education and play at home. Some (the exception) invested a lot of time and effort, hired top musicians to get lessons, and became very good. The children of well doing families would also take drawing and painting lessons.

> did any of these well-bred women ultimately become composers?

If you mean write music for domestic use, yes, it would not be too rare to write some dances or songs.

Now to the longer version. I can only offer a general idea, since I am not terribly into the social aspects of music, or Regency England.

I think you are really asking about women being professional musicians, which is a very different thing, for both men and women in high society. Let's start with what applies to both, and then we get to the specific situation of women.

The conception of musicians as artists, and not just mere servants doing their job, is quite advanced but still being developed during this time. A century before this period musicians were clearly... servants, not particularly relevant people. By the end of the Regency the idea of music being an art and musicians being talented and important is closer to what we have now, but not quite there. In this period musicians were becoming some kind of professionals.

So, would the high-society and nobles want to be professional musicians? No, not at all. Why would they? It was beneath them. Can you imagine a princess wanting to become a cobbler? Yeah, being a musician would not sound much more acceptable than that.

What about landowning and professional families? It sounds less insane, but not all occupations would be on the same level. What would be more appropriate for a respectable family: being a military officer, a successful law professional and politician, or a... musician?

Now, what about women during this period? The women in these families were expected to get a suitable husband, have children, and take care of domestic affairs. They were not public figures, would not have a profession, and were not really supposed to work. This is not compatible with becoming a professional musician, so becoming a famous composer was not really an option.

Could these women work? Yes, if it came to that. An unmarried woman could be some sort of governess or perhaps teach, but for middle and upper class young women this was seen as something below their status, and not all would have sufficient education to teach. These were rather domestic, indoors occupations, working at a small businesses in public was not a thing for these women.

Were there any women with a career in music during this period? Yes, yes there were.

Maria Hester Reynolds (1760-1813) was a pianist, singer, composer, and teacher. She gave recitals on piano and harpsichord before getting married, after that she became a teacher and composer. Her pupils were from the nobility, so she must have been very well regarded as a teacher, and her music was popular in England. She corresponded with F. J. Haydn, who was THE top composer in continental Europe and was extremely well received in England to the end of his life.

Here is a piano sonata by her: https://www.youtube.com/watch?v=yNf2lUN4Ojw

Jane Mary Guest (1762-1846) was a composer and pianist, who studied with J. C. Bach (son of of J. S. Bach, the youngest member of the Bach dynasty of awesome composers). She would give public performances and published music in the 1780s. Later, she taught the daughters of the Royal Family.

Here is a recording of one of her sonatas (which were very popular): https://www.youtube.com/watch?v=bg0-4i-NdqA

I mention their connection to these famous men as a reference, because these women were not from super rich families. Being a student of J. C. Bach at the top of his career, or corresponding and exchanging music manuscripts with Haydn do not sound like average amateur musician.

Stopping completely, or at least reducing a lot their public performances after being married, is something we see in the lives of many women in music.

If you are interested in reading about some other women in music history: https://www.reddit.com/r/AskHistorians/comments/3niee0/it_was_common_for_upperclass_european_women_prior/cvoj4p7/

The lives and careers of Fanny Mendelssohn, Clara Wieck, and Amy Beach would be relevant to your question in terms of the possibilities of a career in music for women during the 19th century. The executive summary: among the few women who had a career in music we see that it helped enormously to be born in a family of musicians, or have some wealth. Their performing career was pretty much over once they married, during most of the 19th century (Clara Wieck is a big exception, because her professional life had pretty much all the highlights any top notch pianist could want today). Their compositions were mostly for piano solo, voice and piano, or small ensembles. It is rather rare to see orchestral music or opera composed by women, because they usually had no way to get the required education, experience, and/or opportunities for this. Amy Beach is the exception here, and I would seriously recommend you to listen to her music.