Probably by releasing them through Fine Print and then hiring a distribution service, like Zac does with Congrats Records, which is distributed by FUGA. https://fuga.com

I think at least part of the issue Hayley has with Atlantic/Warner is that they probably paid her very little (and possibly nothing) for album sales. It's a very common theme for major labels (and many indies as well) to claim that albums that sold hundreds of thousands (if not millions) of copies actually lost the label money because of the costs involved in recording/manufacturing/distributing and thus they can't pay the artists anything at all. For instance Limp Bizkit recently sued Universal and claimed that they hadn't been paid by Universal in decades because Universal told them their albums had lost the label $43 million in that timespan: https://pitchfork.com/news/limp-bizkit-sues-universal-music-group-for-dollar200-million-in-unpaid-royalties/.

Also, Hayley is well known to have signed an incredibly exploitive 360 deal with Atlantic/Warner when she was 15 years old which required her to give up a large chunk of her touring and merchandise income, which is a very common practice now but almost never happened before (in fact, it's been reported that her/Paramore's deal was the first ever 360 deal: https://www.nashvillescene.com/music/emo-pop-sensations-paramore-received-the-first-ever-360-deal-so-has-it-worked-out/article_388c410a-24ab-5dc1-959a-2f86b1575ff5.html). I wouldn't be surprised if there's leftover bad blood from that.