The Lib Dem said safety at the accident blackspot, just over the Peeblesshire border, was not simply the responsbility of Labour-run Midlothian Council, which has been criticised recently for lack of investment in the junction.
Councillor Bhatia told last week's SBC meeting that the two local authorities, along with the police, should work together to mitigate the dangers at the junction of the busy A703 and A701 commuter routes.
However she backed a Conservative call for SBC leader David Parker to write to Midlothian Council "seeking its proposals and plans for improvements to the junction ... as the continuing level of accidents is unacceptable".
In support of his motion, Councillor Neil Calvert (Tweeddale West) said the junction was used by many motorists and bus passengers
from all over Peeblesshire going to and from Penicuik and Edinburgh.
"While some improvements have been made in recent times, these have not significantly reduced accidents," said Mr Calvert. "There are reported and unreported near-misses almost every day and, because of this, we are seeking to apply some pressure to our neighbouring authority (Midlothian Council] to take some further action to reduce the risk.
"Considerable investment would be required to change the junction to a roundabout and this is generally regarded as the best solution. However, it is also recognised further improvements could be made in the meantime which would undoubtedly reduce the accident risk."
Mr Calvert noted that, in the past, the south of Scotland regional transport partnership SEStran had included investment in Leadburn in its programme, but that body's capital budget was no longer ring-fenced and had been disbursed to each of its member local authorities.
"It is important we try to maximise our influence on Midlothian Council and ask it to consider its priorities and invest in improvements to this dangerous section of road," said Mr Calvert.
Councillor Willie Archibald (SNP, Tweeddale West) told the meeting: "This is a nightmare junction which affects motorists from across the Borders. It is the complete responsibility of Midlothian Council to rectify it."
But despite calls for action from national politicians of all parties, Councillor Bhatia struck a more conciliatory note.
"I know only too well the issues surrounding the junction, especially following the death there of an elderly constituent of mine (Janet Spavin, 87, who died after a two-car smash in January], but political mudslinging will not solve anything," said Mrs Bhatia.
"The Leadburn junction is of much more significance to people living in the Borders than it is to those in Midlothian and the regional boundary signs do not mean we can simply wash our hands of it.
"We could, for example, take measures on our approach roads, warning motorists of the junction. While I support this motion, let's be careful not to pass the buck. We must work with Midlothian Council which, like us, is under severe financial constraints."
Councillors unanimously endorsed Mr Parker's letter to Midlothian Council and noted that a site meeting, involving roads officials from both local authorities and the police, was due to take place this week.
A spokesman for Midlothian Council declined to comment on SBC's decision.