Though I'm in Broward, I have a similar situation -- limestone rock just a few inches beneath the soil. Fortunately, the mango tree isn't all that sensitive to alkaline soil, so I've been able to get good success by simply mulching over the entire backyard. It took a couple of years, but somehow the humus helps the tree to absorb micronutrients. My carrie mango, which previously had signs of micro-nutrient deficiency (ie, chlorosis), is now green as grass in the summer without micronutrient application due to the layer of compost in my yard.
However, the same can't be said for my carambola. It requires multiple applications of chelated micronutritional spray multiple times a year, regardless of how much compost it has.
If you don't want to go through the trouble of mulching over your backyard with a thick layer of tree trimmings, then you could just hit it with the southern ag chelated spray that Zands referenced a couple of times per year. Non chelated forms of micronutrients (eg, granular fertilizers) will not work at all if you have alkaline soil.