Is this for commercial purposes you are developing this .. to sell? ... or a hobby undertaking?
You are not going to get around what Retro-techie is pointing out. Based on the size constraints you have shown, without going with custom cells, you are limited to 3500mAH * 2 * 3.7 or about 25.9 watt hours. Assume say 93% boost efficiency and that gives you 24 watt hours at the batteries.
Let's assume you get 200 lumens/watt at the LEDs, and 87.5% optical efficiency. That gives you 175 lumens/watt at the LEDs ..... again best case.
So you have 24 * 175 = 4200 lumen hours .... best case. That assumes not inexpensive batteries AND it assumes high binned LEDs and more of them than typical. Practically you may be more in the 3500-4000 range, but let's take 4000.
You said you would be willing to have it dim from full to 50% over the discharge. That means 75% average brightness, so let's say 5300 total lumen hours based on initial brightness (roughly). So you could start at around 500 lumens, dim down to 250ish, and stay on for 10 hours .... best case.
Of course, here are two things you need to take into account:
- If you are doing this commercially, you better take a look at the patents out there. There are products like this on the market and some have been patented.
- Given the construction you are showing, and depending on bulb orientation (or if in a recessed can), you are going to be cooking your batteries ... and cobalt electrode lithium ions and related types ... don't like to be cooked. Given the way you describe it, the bulb is going to be on most of the time, that means you could be cooking your batteries 24/7. That 4000 lumen hours we calculated above could be 3000 and then 2000 in not that long of a time.