There is no definitive answer to your question, other than to say you're supposed to have your passport on hand or readily accessible if prompted by the authorities but based on many threads/posts on ThaiVisa success seems highly variable. Some seem to get by with a Thai driver's license - for us foreigners these have our passport number on them (for ID Number), while others get by with copies of their passport and visa (or entry stamp). I'm guessing you don't have a Thai DL.

I carry a photocopy of my passport, visa (I have a non-Imm B) and Thai car DL with me at all times.

I guess I wouldn't count on being able to board, but maybe with a copy of your passport, entry stamp page, another quasi-government photo ID from your home country and maybe even something related to your application for an Indian visa in Bangkok, you might improve your odds?

You can Google: domestic flight no passport site:thaivisa.com

and grind through the various threads.