How do I host a mintty shell inside of Console2?

You can not do that.

Console2 (being a "Terminal Emulator") hosts terminal programs .. as in "hey, I will render the chars you want to output and I will give you what the user types into the keyboard". Useful terminal programs mostly used in "Terminal Emulators" are "Shells", such as "cmd.exe" or "powershell.exe" or "bash.exe" or "zsh.exe". If you like it, you could also host "python.exe" from within Console2.

Mintty is also a "Terminal Emulator". For that reason, you can not do what you are asking for.

Just to explain the used 'terms' (hehe) a bit more:

• Terminal: "The function of a terminal is confined to display and input of data ... A text terminal, or often just terminal (sometimes text console) is a serial computer interface for text entry and display." Back in the good old days these "terminals" were really hardware and looked like:

  

• Terminal Emulator: "A terminal emulator, terminal application, term, or tty for short, is a program that emulates a "dumb" video terminal within some other display architecture. Though typically synonymous with a command line shell or text terminal, the term terminal covers all remote terminals, including graphical interfaces."

• Shell: "A shell is a piece of software that provides an interface for users to an operating system..."

You cannot host MinTTY, but you can use directly the Cygwin bash as the Console2 shell. Just create a new tab with \bin\bash.exe --login -i as your shell