Is it possible to name the 'screen' logfile from the -L flag?

You can edit your .screenrc file and add something like this:

logfile /tmp/myownlog

It is also answered at Specifying a log name for screen output without relying on .screenrc.

Probably the easiest way is to use the following (starting from Screen version 4.06.02 and up):

screen -L -Logfile log_filename your_command