At 12 a volt 3 Kw heater is 3000 watts or 250 Amps per hour.
An average leisure battery is 100 Amp Hr, leisure batteries are discharged to about 50% of their capacity, so a 100 Ahr battery would supply only 50 usable amps. Go figure.
By the way a 55 watt bulb ( which is only 4.5 amps ) would flatten it in 11 hours.
The idea is right, but you're a little confused by the units.
You're right about a leisure battery storing about 100 Ah. That is actually the amount of charge that it stores. 1 Ah = 3600 ampere seconds = 3600 coulombs, (Coulomb = the SI unit of electrical charge). It's not a direct measurement of the amount of energy stored. But an approximation would be 4.3 MJ (megajoules).
(That's 12V x 100A x 3600s = 4 320 000 J)
(Amp hours are used as it's a simpler unit for most people to work with. So you can draw 1 A for 100 h or 5 A for 20 h etc. It's a rough and ready method, but it's good enough for non-techies.)
Energy is measured in kWh, kilowatt hours. 1 kWh = 3 600 000 Joules, (Joule = the SI unit of energy). You buy energy from the electricity company in kWh.
Power is the amount of energy delivered every second and is measured in kW = 1000 watts. 1 watt = 1 joule per second. When talking about heaters, it's the power that we're interested in as that tells us how much heat energy is delivered every second. As the watt is quite a small unit, we use kW.
Anything that delivers power is measured in kW or W. Both gas heaters and oil heaters and even wood burners are rated in kW and even your car/van engine. Smaller things like light bulbs and small electric motors are rated in watts. Tiny things like lasers are rated in mW.
(Interestingly, this is only useful when talking about heaters with very low-power electric motors. When talking about large electric motors we talk about kVA instead. But that's another story!)
There are relationships between voltage, current, charge, energy and power, but they are too complex to go into here, and we don't need to know them anyway.
There is, however, no such unit as amps per hour, and a 100Ah battery can supply as many amps as it likes depending on the external and internal resistance although a very high current draw for a long period will damage it.
But the long and the short of it is that you are absolutely correct: a leisure battery could never run a 3kW heater under any circumstances. Leisure batteries are designed to run your caravan TV and low-energy lights, shower pumps etc, not high-power heating circuits.
If you need to heat your van, it must be either using electricity from the mains or a generator or by burning some chemical energy.
