I have my best lifting at lunch time as well. I think being awake for a few hours prior to the workout is definitely best.
How long do you have between PT and the workout? I can see some benefit in terms of the body being activated, getting blood flow with the early workout. But you also say you are "beat down" which doesn't seem conducive to helping the lunch workout. That is very interesting.
James 1:16-17 ESV
Do not be deceived, my beloved brothers. Every good gift and every perfect gift is from above, coming down from the Father of lights
With God's help...Mens sana in corpore sano