Direct transfer by taxi from Mashhad to Bajgiran

I doubt you can book a taxi in advance (though my experience of travelling in Iran is 14 years old...).

I would suggest contacting an Iranian travel agency to organise a minibus.

This agency seems to be recommended on other travel forums:

http://www.irantravelingcenter.com/

This one states it provides transfer services:

http://www.iranparadise.com/en/service

A few others to try:

http://www.letsgoiran.com/

http://www.iransafar.co/

http://www.welcometoiran.com/

If they can't help you directly, i'm sure they'll put you in contact with the right agency. There are more out there!

Iran is amazing, i yearn to go back. Enjoy!