An XHP50B-00-0000- 0D0BJ440E (XHP50.2) configured for 12V should produce 1000 lm at 625 mA. This LED is 4000K, 70 CRI, and currently in stock at a distributor I use often. Vf should be around 11.2V at this current. This is 7W and over 140 lm/W.
With a 4S LiIon Pack, an LDO linear regulator could do this job. 12 A-h/0.625A = 19.2 hours run time. That's an average efficiency of about 80%. You should be able to do better with a switcher. At 90% efficiency I calculate 21.6 hours run time.
Do you need to run this all night for several nights in a row? That's the only reason I can think why I'd want to recharge on the go.
A 4S4P pack with 3000mAh Cells would be about what you are talking about. If it were driving a switching regulator, I would say the average input voltage is 14V, and calculate 14V*12A-h = 168 W-h. 7W at 90% efficiency means 7.8W load on the pack, giving the above stated 21.6 hours.
Because of the lower currents involved, I would consider this a more optimal configuration than trying to run a 3V-ish LED from a 1S16P pack, but that would work too.
Building on what you and others have said, there are several ways to get longer run times.
1. Lower the output.
2. More efficient LEDs. I believe 200lm/W is possible today, giving a 43% increase in run time.
3. Build a bigger battery pack.
4. Have multiple packs - can support personnel carry these to swap out?
5. Recharge your pack.
Since you specifically asked about recharging on the go, let's talk about that.
Your pack is around 168 W-h. This would take about 200 W-h input to recharge from fully depleted (less if not fully depleted). The large difference between 168 and 200 is not because of the battery charge efficiency, which is quite high. It's because of the efficiency of the charger, which isn't likely to be quite as high.
Given the battery life numbers, if you could charge every day you would only need to charge to around 70%.
If you had access to a sufficiently powerful charger, you should be able to do 70% in about an hour. This would take 120-140W. With appropriate preparation, you could get this from a 12V vehicle battery. Just plugging into a DC outlet in a vehicle, you should probably plan on a good bit less than this, and maybe 2 hour charge. With line power, 1-hour charge to 70% is easy.
A 50W solar panel could provide enough power to fully charge in 4 hours, 70% in 2.8 hours. A 20W panel running all day could probably keep the battery charged, but even this is pretty big to carry on a bike. Unless you are talking about many days, more batteries is probably cheaper, smaller, and lighter.
Wheel-powered generators have been mentioned. I haven't looked at these in decades. What I remember from back when was so inefficient that I wouldn't even consider it. If modern versions are available that are well-engineered and well-integrated into the bike, this could be a solution. You'd need up to 20W for 10 hours depending on how much charge your battery needs. But even this small a load, even if pretty efficiently drawn from the wheel, would represent a substantial increase in drag on the bike. I think it would be far easier to carry 5 times the battery than it would be to charge one off the wheel.
In summary, I'd say that the only way I'd consider trying to recharge batteries on the go is if there are no support personnel or vehicles along your route, high light output is needed, and there's a need to run the lamps 50+ hours on your trip. Otherwise, I'd carry sufficient batteries, or swap them out along the way.
