A friend used a mobile crane hire earlier this year (to move his
post&beam garage intact). The hire company quoted 2 prices, one to do
the lift as you instruct (and you take responsibility), and a second
rate they will take charge of the lift and ensure it's safe
completion.

Getting the crane company to do any over-the-house blind lift (i.e.
the crane operator depends on remote instruction during the
positioning) may be the answer to having the whole problem sorted in
one go.