I go with carbonfiberboy. Find a pace that you can maintain up the hill and stay with it. If you need to, put a bigger cog on your rear cluster. The trick is to go up no higher then your anaerobic threshold and stay there. Once you cross it, you are living on borrowed time. Unless the top of the hill is quite close, you will have to stop. But it is your choice. If you like red lining it until you have to stop that is your business. It is just another form of interval. Just recover however much you want and try again.
I have outclimbed quite a few people who blew by me and blew up before they reached the top.
Also, about stressing your heart muscle, hearts have their own muscle type --- cardiac muscle. So they really are not like skeletal muscles. I have seen a number of cyclists go hard enough to pass out. That is a result of the blood being shunted to the muscles leaving an inadequate supply to the brain. I have yet to see a fatality from a heart attack on a bike ride and I have never even heard of one. So you are probably reasonably safe.