Honestly, I try to avoid riding my bike whenever I'm on my period. But if there are times that it's inevitable, I'd use an overnight pad just so I can feel secured. Yes, it can be thick and uncomfortable but I'd rather have that than have any leaks. As far as clothing goes, I just wear thicker shorts or jogging pants. Leggings are great as well, make sure they're on the thicker side. Anything black is advisable so in case of any leaks, it won't be as noticeable.
Overall, you can go ahead and ride during your period. It's actually a great exercise to relieve cramps, if you get them. Just make sure to bring an extra pad and underwear with you. I hate tampons as well so I don't recommend them. Once your period lighten, you can use the thinner pads to make sure that you're still protected.
Right now, I'm on contraceptive injection though so I rarely get my period. And even if I do, it's very light and doesn't need pads. You can have that option as well although it's not really required.
I hope this helps!