It's easy to get amex variant, just DM ICICI on X and ask for amex variant.
They will call you just tell that you need coral amex LTF against coral rupay LTF.
1st scenario
They will simply give.
2nd scenario
They will say that they can only issue Visa as they only have that option bla bla. Just tell them that your friends and family members hold all 4 variants that too LTF so it's possible and you can share email proof as well thn they will say we will check with our concerned team and let you know, after sometime they will call and say we can give you amex variant.
Ezzz
Don't apply all variant together. Do it one by one and take visa at last coz that's easiest.