If I copy a bpmn file (it's an XML-based markup text) under /public, apparently, I get binary content in my browser. The content is returned with the type application/octet-stream.

It is probably expected because the library used by SWS cannot guess that extension so it can map its content type. In that situation, SWS falls back to application/octet-stream as a "safe-bed".

Can I provide a configuration to map this specific extension to be served as text/plain or anything that won't mess up the character stream?

Yes, you can by using Custom HTTP Headers via a config file. Like this example.

my-config.toml

[general]
root = "/public"

[advanced]

[[advanced.headers]]
source = "**/*.bpmn"
headers = { Content-Type = "text/plain" }

And run SWS like static-web-server -w /my-config.toml