As of 8 April 10am
ATO Small Business Newsroom - article is wrong
The article about COVID-19 and the statement that
"Make sure you lodge your March and quarter 3 activity statements on their due date and no earlier."
IS WRONG. It will be updated soon.
You can definitely lodge the March BAS as soon as you are ready. Earlier lodgement will not impact your eligibility for the stimulus package.
Other parts of the ATO website confirm this as does the Treasury information