Mansion House to Mornington Crescent is possible in exactly five moves if LV>6.4 and Bank Holiday rules apply. I have a really simple proof for this, but my heart isn't what it waaaaaa....
[Blob] I'm not sure Bank is useful here - you might have missed a negative somewhere, as I don't find anything useful in that avenue... Try applying triple chain rule, and you'll find that 3.Willesden could be more helpful... I haven't followed it through yet though.
To clarify, Mansion House is move zero, aiming for MC at move five. Anyway, I shall try 4. Lancaster Gate, with negative spin. Is it possible from here?