It is different if you have an athletic base from power lifting and running, and lots of skill because of your love for the arts, as well as sparring and rolling.
Running and weight lifting are usually at a different time of day from class. 90% of martial arts is at a low intensity - drills, shadow boxing or working on single techniques and partner drills, and learning grappling moves.
Like 10% is hard: rounds of standup grappling or sparring rounds with shots that can damage the body.
Imagine a three hour training day:
30 minute run, 30 minute weight lifting or fitness training, 9 hour break, 2.5 hours partner drills and technique, 8 rounds rolling or sparring. Of those 8 rounds, half should be light, minimum. Hard impact once a week is enough.
Edit: I'm not a fighter or trainer, and hopefully someone who is posts something better. My point is mostly that professionals spend WAY more of their time at low levels of output, improving their skills and targeting their fitness goals.
The gyms where common people train aren't really cut out for this. Most BJJ and Kick Boxing classes are half workout and little technique, little critique. It is hard to run and lift heavy while hitting group classes 6 days a week, because they can be so exhausting physically and so damaging to the body - from hard sparring and countless push-ups.
If you don't have a great trainer you can trust to guide you, then you basically just want to spend as much time out of class mastering striking technique on the bag and on the mirror, get private grappling lessons or open mat time where you can ask questions, and prioritize getting into great shape through losing fat or power lifting or both. If all that means less group classes, so be it.
