I was in a similar situation as you (south Indian with only one name), and after trying to reason with their customer support, I found it easier to just use a fake name.
If you think my real name is fake for no good reason, go ahead and have a fake name that "sounds real".
(Not sure if you'll see this reply this late, but...)
The situation regarding ID's is a mess, but I believe I'll be able to fly, to the US or wherever, without issues (though I haven't done so yet).
The reason is that, India being a multicultural country, our central government that issues passports assumes a firstname-lastname pattern as well. And back when I got the passport, I didn't yet understand the firstname-lastname concept, and just followed the passport office employee's instruction: "don't leave that 'surname' field empty, that's for your father's name". I was curious why the heck someone would label that field as surname, but just took it as part of bureaucracy. And so, I have a passport to the name "myname fathersname".
The "mess" I referred to in the first line is about how different ID's have my name in different forms: let's say my name was David and my dad's is Michael. I have government issued id's that say my name is:
