Tired brain + workouts

I work as a copywriter and some days (over the last few months more often than not) its very intense - up to 10hours of constant creativity and precise focus on details. Sometimes I will get to the end of the day and I'll feel like my brain is totally drained... I try to workout 4/5x weekly but when my brain is super tired like that, I feel like I'm struggling to maintain form and even that my balance isnt as good as when I'm rested. I don't want to skip too many sessions but I feel wary that I could hurt myself...

I have been criticised before for not having a 'real job' and honestly dont know whether this feeling "brain fatigue" is a real thing and if it would impact my physical performance as I feel it does or if I'm inventing that?

Anyway, I would love to hear opinions and if someone has experienced anything similar. I'd also love workout suggestions that would be safe but still allow me to get some decent movement in.

Edited