Using LED bike lights with a dynamo instead of batteries
LEDs need to have current limiting to work properly, so your system as shown would blow up the LED in short order.
You also wish to have some extra hold-up time for your system, which implies a secondary battery. This needs a battery charge control to prevent overcharging your battery. Again, hooking up to the unregulated rectified dynamo is a bad idea, you could damage or explode your battery.
Instead, my thought would be to use a USB-chargeable LED light, and convert your dynamo output to regulated 5V DC to charge the light while you’re riding. Then you have a pre-engineered solution for your lamp and a much simpler problem to solve for your dynamo. In fact, this is a product you can buy pre-made. Bonus: charge your phone too.