Gymnastics is the best way to build functional strength.
Weight training does nothing for improving power and speed in martial arts. However, moderate weight training does have a place in building stabilizer muscles used for martial arts. Squatting and deadlifting your body weight should be enough for a martial artist. Anymore than that and it starts to hamper your martial arts training.
The lower back doesn’t need to be strong (as in ability to lift crazy amounts of deadlifts) but it needs to be stable.