Here's an update on how I'm progressing with this. I took the second approach from my initial post and added the following code to my script.

#This is a function to override SQLites DATETIME('NOW').

sub my_fake_time {

    # gmtime() normally returns the current time but it is overriden b
+y Test::MockTime to give a fake time.
    # Actually I am using the version of gmtime() from Time::gmtime as
+ it has a nicer interface.  As this too overrides 
    # Perls core gmtime() function it must be use'd after Test::MockTi
+me so all the overrides occur in the right order.
    my $t = gmtime;

    # return the fake date in the same format as SQLite returns.
    return sprintf '%04d-%02d-%02d %02d:%02d:%02d', $t->year + 1900,
        $t->mon + 1, $t->mday, $t->hour, $t->min, $t->sec;

};

...

# This works.  DATETIME('NOW') in my SQL now returns a fake time.  

$dbh->sqlite_create_function(
    'DATETIME', 1, \&my_fake_time
);

# However CURRENT TIMESTAMP still doesn't work even though from what I
+'ve read it is implemented in terms of DATETIME('NOW') 
#
# I tried this but it doesn't work.

$dbh->sqlite_create_function(
    'CURRENT TIMESTAMP', 1, \&my_fake_time
);

# Any other ideas?
[download]