Farron, who represents Westmorland and Lonsdale, raised concerns in Parliament following the announcement of a strategic review into how the Post Office operates, Local Democracy Reporting Service said.
It revealed it could get rid of 115 directly owned branches, which also includes Furness House in Barrow.
Farron said the same branch was threatened with closure three years ago.
“We won our campaign to save it, in part because Post Office Ltd could see there was not space at WH Smiths in Kendal to take the Post Office, that has not changed,” he said.
The campaign also argued Royal Mail could lose its sorting office as a consequence of a closure.
“What has changed is that two more high street banks, Halifax and Lloyds, have deserted Kendal town centre on the basis they claim that we’ve got the Post Office down the road, and they will be able to take up the slack,” Farron said.
Halifax and Lloyds are due to close in February.
There are still a number of banks branches in the town, including Barclays and HSBC.
In response, Thomas said: “There are significant additional costs associated with the directly managed branches and it’s right the Post Office looks at those questions as well.”