However, direct election of US Senators did not occur until the Wilson administration, I believe. That is after recall laws were enacted. Who represents a state in the house or senate are subject to state law jurisdiction. That is why some states allow a governor to appoint a Senator to fill an empty seat and other states require a special election.
I say start the damn petition. If LA law recognizes the recall and an election sends a replacement, it would pit the US Senate against the voice of the people. That would make great press. I bet the Dems would back down.
And if nothing else, starting the drive will cause Landrieu to incur legal costs and start looking over her shoulder. She may become reluctant to sell herself on future votes. It will drain her campaign funds and weaken her as a candidate next election.
Some states are variable, such as Massachusetts :)