I am going to take a wild guess ... since you have goten no answer.

Set up a awstats install 'mutiple_sites'
and in the configuration file point the log's variable to

my @multiple_logs = ( 'log1.log', 'log2.log','log3.log');
my $log_path = @multiple_logs;

In other words, create an array of the log files then rename that array to the original variable name.

Might work, never tried to make a $string from an @array of log file lines ...

Good coding experiment anyway :P