Bechard charges that Broidy's refusal to pay is a breach of the agreement. She is asking for the remainder of the payments, $1.2 million.
Broidy, a Los Angeles-based investor who served as the RNC's deputy national finance chairman, resigned in April after details of the affair were made public. He acknowledged a "consensual relationship" with the woman and called all the publicity "unfortunate."
In the lawsuit, Bechard says she first met Davidson in October 2017 when she was seven weeks pregnant with Broidy's child. Details of what was discussed in their meeting were redacted in the lawsuit.
Davidson also represented Daniels and Karen McDougal, a Playboy model who also claims to have had an affair with Trump. Both women have accused Davidson of working with Cohen, the president's former lawyer and fixer, to silence them and create unfair agreements that catered to Trump.
Bechard also claims that unbeknownst to her, Davidson was working with Cohen and he was "recruited" to "solve" Broidy's problem, according to the lawsuit.
Cohen's business practices, including his role in the hush agreements, is now part of a federal investigation.
Bechard says Davidson did not accurately portray details of the agreement she signed in December, including not allowing her to have a copy, and took a 35 percent cut of all the payments she received.
"Despite his fiduciary and ethical duties to his client, Mr. Davidson treated Ms. Bechard's claims as a commodity to be traded for his own financial gain," the lawsuit says.
She says $1.6 million was the agreed-upon amount because it was the total of what Broidy would have had to pay in child support.
The complaint says Davidson told her the agreement and money wasn't in exchange "to have an abortion but rather to give up her rights to sue Mr. Broidy and to not to talk about the relationship."
After signing the agreement, Bechard did have an abortion, she says in the lawsuit.
She claims she did not want details of the affair released but her name, the pregnancy and the agreement were publicized after Davidson disclosed the deal to Avenatti.
Avenatti tweeted some details of the deal and, the following day, the Wall Street Journal included Bechard's name in a story about the affair and agreement.