If you have any LTF gemstone card, then just mesg them on twitter they will call you and give a LTF Rupay Card to you.
But as you said you don't have any LTF Core Card (ignore APAY), chances are a little less.
But also since your limit is low, try sending your income docs to
headservicequality@icicibank.com
They will increase ur limit manyfold.
After LE you can request for card upgrade.