You are right. After eating, your body prioritizes digestion and the blood flow is concentrated at the core instead of the brain. This, of course, has a negative effect on the performance of any task done, when compared to the same in an empty stomach.
An interesting take on this is reported here (paywall). In a nutshell, this particular study assumes that the brain might perform better on an empty stomach as humans characteristically have evolved to have a focused brain for our inherent need to acquire food. In mildly starved mice, the hunger hormone (ghrelin) seemed to affect their cognitive abilities.
On the other hand, consider psychological factors too. There is an inherent connection between your gut and the brain. Emotions of an individual have a greater effect on their performances. If you are hungry, you might feel anxious. This change in the mood can adversely affect your performance too.
But maybe some people would play well under anxious and stressful conditions. And some may not. And after a full meal, some may find it relaxing and they perform better.
I am afraid, I don’t have a definitive answer. I play Battlefield (not professionally though). Personally, I’d prefer to play in a relaxed mood after eating or when I am mildly hungry. Maybe you can experiment with full, mildly starving, and empty stomach situations and find the right strategy.
