I'm not sure why you are chomping unless you are doing more with the data than you have posted. Also, you are capturing then not using it. Once you have your "L1" or "L3" put the "zero or more of anything" inside the negative look-ahead and make it non-greedy.

use strict;
use warnings;

while ( <DATA> )
{
    print if m{^L[13](?!.*?TEXT)};
}

__DATA__
L1 04:10:07.915  LOG:   Want this Line1
L3 04:10:07.915  LOG:   Want this Line2
L1 04:10:08.024  LOG:   TEXT. Do not want this Line3
L3 06:37:58.163  LOG:   TEXT. Do not want this Line4
L3 07:02:36.921  LOG:   Want this Line5
L4 08:02:30.910  LOG:   Do not want this Line6
L5 08:02:36.943  LOG:   Do not want this Line7
L6 09:02:38.811  LOG:   Do not want this Line8
[download]

Produces

L1 04:10:07.915  LOG:   Want this Line1
L3 04:10:07.915  LOG:   Want this Line2
L3 07:02:36.921  LOG:   Want this Line5
[download]

I hope this is helpful.