Bernstein, 73, was one of five former executives who last year signed a letter that described the world's oldest soccer federation as an outdated institution held back by "elderly white men".
British Sports Minister Tracey Crouch warned in February that the FA was in danger of losing 30 million pounds ($37.29 million) of public funding and that legislation could be brought in to force change.
The FA subsequently proposed reducing its board from 12 to 10 members and increasing the number of women on it to three by 2018.
Nobody will be allowed to serve more than three terms of three years and 11 new members will be added to the 120-strong FA Council "to ensure it better reflects the inclusive and diverse nature of English football".
"I don’t think they have done enough," Bernstein, who was FA chairman between 2011 to 2013, told Reuters at a Cogress Investor Club event.
"I think there’ll be enough to get the government to go along with it because they want an easy time, want to get it settled, but it won’t be enough to make a real change," he added.
"The board needs to be independent, not controlled by vested interests...and the FA council needs an axe taking to it, frankly."
Anti-discrimination group Kick It Out has described the proposed reforms as "a sham" that still failed to give sufficient representation to minority groups.
The FA has scheduled a formal vote for April 3. If majority approved, the reforms will be voted on by shareholders at an annual general meeting on May 1.
"The danger is that the work that has been done will get accepted just to get rid of the issue because the government’s got enough on its plate with Brexit and so on," said Bernstein.
"And then we’ll be in a very poor period for the next 20 years. I think we’ll have missed a real opportunity.
"Given the fact that we’ve had a select committee report, a parliamentary debate and a vote of no confidence in the FA and a minister who says she wants to see real action, we’ll never get this opportunity again.
"If it ends up with a fudged compromise, then it’s a missed opportunity."
(Editing by Ed Osmond)
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