'DAB leaders blamed their defeat largely on the party's support of an anti-subversion bill.'
The adverb 'largely' has nothing to do with size. It is used to introduce the main reason for a particular situation. Example: He was convicted largely on the DNA evidence.
You may replace it with 'chiefly' or 'mainly'.
The phrase 'at large' has two meanings.
It refers to something as a group or one that is moving about freely. Example: The government must understand what the public at large want before it introduces any reforms to the education system.
A criminal at large is not under control and may cause harm. Example: The killer is still at large.