How to change the default “save” folder in Mutt

You could pipe the message to ripmime, which lets you specify a destination directory on the command line...e.g.:

ripmime -i - -d ~/myfiles/

The -i - tells ripmime to read from standard input, and -d specifies a destination directory. If that's too much typing, stick it in a shell script and call that (which will also make it easier if you want to use other of ripmime's many options for controlling its behavior).

If you add a macro like this to your configuration:

macro index,pager X "<pipe-message>ripmime -i - -d "

Then you can press "X", enter a destination directory, and press <RETURN> to run the command.

OfflineIMAP and isync are both programs that integrate well with mutt and will satisfy your other criteria.

OfflineIMAP is written in Python and isync in C; both are very quick.

Both programs are well documented and straightforward to set up; isync perhaps slightly easier.

There is one significant difference between the functionality of each that is worth bearing in mind: you can use mbsync (isync's executable) to repopulate a remote IMAP maildir¹ from a local copy, OfflineIMAP cannot do this. [Amendment: from v6.4.0, OfflineIMAP has the createfolders option to create folders on the remote repository].

The Arch Wiki has pages on both, with example configurations and use cases.²