Bain't = be not

But I've always thought that "ain't" is originally an Americanism.

Dictionary.com, quoting Harper's Etymology dictionary, doesn't think so:

1706, originally a contraction of am not, and in proper use with that sense until it began to be used as a generic contraction for are not, is not, etc., in early 19c. Cockney dialect of London; popularized by representations of this in Dickens, etc., which led to the word being banished from correct English.

Wikipedia has an interesting page on the contraction:

Ain't has been called "the most stigmatized word in the language",[24] as well as "the most powerful social marker" in English.[25] It is a prominent example in English of a shibboleth – a word used to determine inclusion in, or exclusion from, a group.[24]

Historically, this was not the case. For most of its history, ain't was acceptable across many social and regional contexts. Throughout the 17th, 18th, and 19th centuries, ain't and its predecessors were part of normal usage for both educated and uneducated English speakers, and was found in the correspondence and fiction of, among others, Jonathan Swift, Lord Byron, Henry Fielding, and George Eliot.[26]

You had already found a source (Wiktionary) which states that

"bain't" is the contracted form of "be not" and it is a British dialect.

Which leaves your question:

the question is "Aren't we full already?". In what part of Britain are we likely to hear "bain't"?

With the (wikipedia) information on ain't (is not) in mind, and the references like yourdictionary for bain't (be not) I would expect bain't to be of similar origin as ain't, and associate it with Cockney (London) English.

However, I am not sure that you are likely to hear it today - contrary to the widely-used ain't, bain't doesn't seem to be used a lot and even online references are sparse and not giving many details.

Very SW England + West Midlands. Though can be found all over southern England. Basically I be-ain't. Sometimes be-ant, or often bist be-est. So a greeting in Bristol would be How bist? Ah b'aint so bad. There be some classic dialect on here https://youtu.be/iPi5LkLhzr8.
I think the Belfast quote is from a popular BBC radio show from back in the day that lampoon's the rural stereo type of SW England called Rambling Syd Rumpo. Well worth a look if your into maximum innuendo poetry. https://youtu.be/hKdJZbRParA Dialects are flying out on England but this video shows there's plenty of pirate Shakespearean dialect still in use is southern rural communities, interesting to compare to Irish, Canadian maritime and American dialects. https://youtu.be/WjTIFkWJctY