The burden Mayor Jane Byrne assumed with her dramatic endorsement of Sen. Edward Kennedy four months ago was brought home in secret city hall meetings on Chicago's tangled finances when one local banker suggested the city might consider a federal bailout, New York-style.

"Are you kidding?" snapped one of Byrne's aides. He explained to the banker that with the mayor committed to pushing Jimmy Carter out of the White House, precious little help from Uncle Sam was possible. On the contrary, Byrne insists President Carter is out to get her. She readily agrees that her multiple troubles began with the Kennedy endorsement.

The reality in Chicago is that Byrne made a different job far tougher by embracing Kennedy's quickly deteriorating candidacy. That contradicts the perception in Washington that Kennedy's illinois problems began with the embattled mayor's endorsement. She, in fact, is the victim.

What's more, her chances for political revival here are much better than Kennedy's. She will have to twist arms of Democratic precinct captains, who disappove of Kennedy on moral rather than ideological grounds, to back him in the March 18 primary. But she may be regaining her own popularity with persistent toughness, such as refusing currently to yield to the firemen's union.

Byrne, surprise winner for mayor in 1979 against the Chicago Democratic organization, was still in a love affair with Chicagoans last Oct. 27 when she discarded a seeming commitment to Carter by endorsing Kennedy. All manner of misfortunes soon followed -- including walkouts of transit workers, teachers and firemen -- and her popularity dropped.

"You just don't treat a president that way." contends a liberal Democrat with no love for either Byrne or Carter. He shares a widely held view that Chicagoans so resented the mayor's switch to Kennedy that they turned against them both. More plausibly, she was dragged down by anti-Kennedy sentiment that began building nationwide just as she endorsed him.

Byrne has a considerably different theory. At the heart of the firemen's crisis, with television camera crews camped outside her fifth-floor city hall office, the mayor quietly and calmly told us that "the administration is using intimidation" against the city because of her.

Besides the Justice Department's desegregation lawsuit, that includes alleged withholding of federal funds and a freeze on the city's proposed new jetport. Her most recent exhibit of federal provocation: a snap federal inspection of emergency operations of O'Hare inspection of emergency opertions at O'Hare International Airport Feb. 13. While federal officials in Washington describe it as a check to determine whether the fire strike had reduced airport security, the mayor thinks otherwise. "They were trying to close down O'Hare," she told us.

Yet Byrne admits no regrets for having jumped to Kennedy. "I thought he [Carter] was doing a poor job for the older cities," she told us, adding that she still sees the president as a loser in November if nominated.

Although Byrne blames Carter for slighting Chicago on funds, she told us that her fiscal austerity will put the city in good shape "with no federal help needed" -- certainly not the New York-style bail-out suggested by the Chicago banker. Fiscal experts familiar with both cities believe that had then-mayor Abraham Beame taken the tax-raising/budget-reducing course followed by Byrne, New York would not have had to go begging to Uncle Sam.

Not even Byrne's most vituperative enemies deny she inherited a fiscal mess from a generation of spend-now, worry-later under the fabled Mayor Richard J. Daley. She has not responded with the familiar pattern of her comtemporaries who surrender to municipal employee unions and then go hat-in-hand to Washington for money: she is one mayor who has stood up to the unions.

That is the core of the impasse over the fire department which Byrne told us "has been a government within a government" that she now intends to control. Not only independent aldermen (such as Martin Oberman, who call Byrne "a pathological lair") but regular Democratic aldermen would much prefer that the mayor back down and settle with the union. But with enough firemen now at work to protect them, the public probably favors Byrne's tough stand.

That is one reason she may revive politically once she sheds the Kennedy albatross. That would be bad news at the Carter White House, renowned for vindictiveness that lends some credence to Byrne's vendetta charges. But in this city, vindictiveness is also a synonym for Jane Byrne. Whatever happens to Teddy, the president might be well advised to make peace with Jane.