Mr. Greenblatt and Mr. Berlanti were two of the Hollywood power brokers whom Ms. Ellis invited to attend a kickoff for the group’s amendment initiative in December. The event was held at a mansion in Los Angeles. Glaad staff handed out hardcover copies of the Constitution, including the 27 amendments. On the final page was Glaad’s proposed amendment, which would also protect women, people of color and disabled individuals from discrimination.

Ms. Ellis started her pitch by citing alarming statistics. Gay and transgender people can still be legally fired from their jobs simply for their orientation in more than half the states. She said roughly 300 anti-L.G.B.T. bills had been put forward in state legislatures since 2016.

“Build this into your scripts,” she told the Hollywood gathering. “You create culture, and we need to build awareness that we’re not protected, we’re not safe — that we need this ultimate protection.”

The next morning, Ms. Ellis flew to Washington to meet with lawmakers about the amendment, which would require approval from both houses of Congress (each by a two-thirds majority) and ratification by at least 38 states.

“I expected some resistance, some ‘you guys are out of your minds,’ but our meetings on the Hill went phenomenally well,” she said, noting that Glaad had hired the Raben Group, a lobbying firm, to help with the effort. “We see a path.”

Glaad calls it the Equality Amendment. It is different than the Equal Rights Amendment, which focuses more narrowly on gender equality and was approved by Congress in 1972; state ratification failed in 1982, although supporters have recently revived that effort. (Glaad is working with the ERA Coalition, which first brought the new amendment concept to the organization in 2016.)