The same headlight bulb running brighter (because of higher voltage through the relay, bypassing the voltage drop in the wiring harness) will use more watts than before.
Switching all other lighting to LEDs will give you more available headlight power, but not a lot. The tail light, running lights up front, and the instrument lights are on "all the time", and switching to LEDs will drop the load by approximately the wattage of all those bulbs put together, probably about 20W total. The turn signals don't matter much at all since they're only on occaionally.
Be careful with tail light LED conversion bulbs. They are usually pathetically dim and unsafe for daytime use - the stop light is not noticeable in sunlight. There are converions using a single 3W or 5W LED that are bright enough, I haven't found a multi-LED "bulb" that's any good (and I have a box of them that I tried).
Changing the turn signal lamps will require a new flasher that works with the low bulb power load, or "resistor packs" that negate any power saving from the LED swap.