I too am allergic to mango sap and have broken out quite badly about 3 or 4 times in the past two years... first few because I didn't know the cause and the last time because I too now have a mango tree (two acutally) in my yard which I got the reaction from trimming the tree, rather than touching mangoes this time... but that was really my own silly fault!
If I were you, I certainly wouldn't cut down the tree. Just try to be careful and avoid touching the sap as much as possible. If you do touch it accidentally, quickly wash your hands with soap and water before touching any other part of your body or face. I've been careful since the last time, and haven't had any problems, even with two huge mango trees and loads of fallen mangoes in my yard!
The mango sap contains the same thing that makes poison ivy and poison oak poisonous, just in a lower dose, therefore it only affects about 4% of people (usually the same people who get really bad reactions to poison ivy). The fruit itself is harmless.... which is good because this way I can still drink mango smoothies, but have someone else to all the work of peeling and cutting up the fruit!