watts=volts*amps
ohms doesn't really matter for this
Basically you have a max drive current of 3a and a max voltage of 5v which is a max consumption of 15w of power.
Typical draw says it's going to be 3.1v at 1.5a, so 4.65w which will give you roughly 500lumens. That will roughly double if it is being driven at max power to get 1000 lumen out of it, and with some efficiency losses, assume it is going to consume 10w's.
The size of your battery is typically quoted in amp hours, which is is the 1500mah and 4500mah that you mentioned above. That means that they can provide 1500mah at 3.7v *assuming lithium*, for 1 hour, or 5.55wh. Since we know the light is going to draw 10w's, that means it can drive the light for 5.55/10 or .55hours. A 4.5ah, or 4500mah battery will drive it for roughly 3x as long or 1.65 hours.
Now, a 1.5v battery with 4500mah, has the same power as a 4.5v battery with 1500mah, because 1.5*4.5=4.5*1.5
So, what you provided is the LED specifications, not the head lamp specifications. This is important in evaluating battery compatibility, particularly from a voltage standpoint as the drivers will be set up to accept different kinds of voltages. If there is a buck driver in the LED board, it is expecting a higher voltage, and if you put a lower voltage in, it won't work or could break something. Conversely if a boost driver is in the unit and you put a higher voltage battery in there, it will most likely fry it. This is why knowing what the drivers are expecting for voltage is much more important than the actual LED specifications.
@Bobby at Under Water Light Dude does a great job at providing real specifications on his lights, so we'll use my current dive light as an example
3500 Lumen Primary Light
Burn Times
If we go into the burn time chart, we find that to produce 3500 lumens of light, the light will draw 30w of power. That comes to 117 lumen/watt for that light. It is the most efficient dive light by a long shot, so don't expect those kind of numbers from most manufacturers. 100lm/watt is about as good as most people will get due to using off the shelf parts. There is nothing wrong with this btw, it's an expense justification that most can't make.
Anyway, so we know that the light is going to pull 30w to produce max power, and we know that it uses a nominal 18v battery pack. What we now need to determine is how long we need that light to run for. I'm a cave diver, so I like to use the rule of thirds on my lights burn time. This not only gives me a buffer to where my primary will last theoretically as long as my gas supply, but since Li-Ion batteries do much better when only used down to about 20-30% state of charge, it's better for the batteries. My current dives are averaging about 3 hours right now, so I need 4.5 hours out of that light. 4.5hours*30w=135 watt hours. Since we know the voltage is nominal 18v, to find out the amp capacity, we divide 135 watt hours/18v=7.5a.
For you, you need to know what kind of battery the light head was designed for in terms of voltage, how many watts it draws, and how long you need the light to burn for. Only then can you determine how many watts you need for a battery and subsequently amperage.
