Be careful with neem oil, on mango the new leaves will be burnt, especially if you put them in sun.
Bordeaux mixture is easily found at least in Europe, try to search for "copper fungicide" in US.
Thiophanate-methyl is the best for mango treatment, a systemic fungicide and it stays inside the plant up to 6 months and the plant really grows and sets fruit well. I would not suggest you this chemical as far as I can see you are not prepared to use harsh pesticides. You need proper equipment to use it as it is hazardous, also it leaves a toxic residue. This is used on tropical plantations, and most likely the mangoes in supermarkets were treated with this. It requires proper timing and precautions.