Should you weight lift? Will it help you be better? Will tightness or bulk get in the way? Sure, yes, and no. Weight lifting is a great way to build and keep muscle tone. I used to lift a lot; I did it in high school, and then picked it back up again in my thirties, and I got a personal trainer. My trainer was a nut about me keeping a log of what I lifted and what I ate… I did not like it very much, but I liked the results. My goal was to build bulk and it worked. Today I still lift, but in a different way. I got myself a Total Gym, yup me and Chuck Norris having at it. Urrgh! Did you see that last rep, Chuck?! Yeah! The Total Gym fits my world right now. I also use Kettle Bells, Chishii (the traditional Okinawan stone levers.) and an old bicycle tube for some judo techniques.
Look, people argue about whether weights will help your martial arts technique and the answer is yes, by creating muscle tone, body awareness, and general health. If you plan to use them to create more power, to be more powerful, exclusively, you are setting yourself up for a fall, because technique is the king, ask Randy Couture. You cannot outrun father time, as strength will fade.
If you bike ride, hit the gym, lift weights, do yoga, rock climb, it is all good. These all increase the quality of your life and hey, have fun.