FKA Twigs was told early on by an indie band to leave the past behind and pick a new musical moniker ... according to emails the band says it sent back and forth with her in 2013.

Here's the deal ... back in March, FKA sued the band The Twigs -- an indie group formed by sisters Laura and Linda Good in 1994 -- claiming they'd sent her a series of cease and desist letters demanding she either stop using her stage name or pay them more than $1 million in compensation.

The Twigs fired off a countersuit against FKA for trademark infringement last week ... claiming, over the last few years, the star has stopped using "FKA" and simply gone by "Twigs."

The groups claims FKA Twigs has "intentionally used her celebrity and resulting power with the media to act in ways designed to increase the public’s association of Barnett and her musical services with ‘Twigs,’ while eroding and overwhelming counterclaim-plaintiffs’ goodwill ... in those same musical channels of commerce.”

They included emails in the countersuit they say they exchanged with FKA back in 2013 ... in which the person they say FKA offered them $15K to allow her to go by "Twigs" while they retain the name "The Twigs."

The Twigs turned the money down ... writing in the emails they are responsible for defending their trademark -- that's the law in the United States. They say marketplace confusion -- artists looking for one band and finding another -- is a massive issue, and they can't make an exception to the rule for her.

FKA's alleged emails show her laying out the importance of the name to her ... writing she created the FKA Twigs persona at a time when she was "very alone and incredibly vulnerable."

While the Goods empathize with FKA in the emails, they're clear -- they're not giving up their trademark ... and they write, "It sounds like you've come a long way, and maybe it's time to embrace a new name -- one that represents where you're going, instead of where you've been."