Wow!
Great find. I love the old chronograph watches. IMHO I would have it restored and wear it with pride.It is like wearing a piece of history. The feel and workmanship of these mechanical devices is amazing.I have my dads wrist watches,my grandpas pocket watch and my great granddads pocket watch.
Based on personal experience,I wouldn't take it your local shop to have it worked on. They have probably never even seen a Tutima. You need to send it to a specialist.
Doug gave some good advice. My thought is to post your request at the Heuer Forum.
I like to mess about with the old Heuer watches. Great quality and lots of history.
The website is
http://www.onthedash.com/
The knowledgeable folks there could give you some ideas on where to send it for a quality service. It wont be cheap,but it will be a worthwhile investment.
Bob