This is extremely disheartening to hear. I don't think Big Basket should have sent the Cease & Desist letter. It seems they have trademarked 'BigBasket' and not 'Basket' (which obviously can not be trademarked). Additionally I also think what the precedents are, for instance did Flipkart actually send C&D letters to lenskart, lendingkart, etc. Please send me private message and I will provide a BigBasket's board members email id and phone number. Good luck to this case and to your new venture :)

P.S Although I think that you should not have chosen the suffix 'basket'. You have gone through extreme lengths to make sure the UI, apps look different. But in general word plagiarism is frowned upon (but it doesn't warrant a C&D definitely). It could have been on the lines of 'letsdaily.in, or letsdailyorder.com', etc