All you are aiming to do is make a portable temporary mains electricity supply so you can use mains operated lights.
Battery - bigger ones will keep the power flowing for longer. Deep discharge types are preferable to normal car types as they will withstand going completely empty more times than standard car ones. Having said that, you are probably still looking at years of use on a standard one.
Inverter - this is the box that converts your battery voltage to 115 / 230V. Usually it will have two thick wires with clips to hook up to the battery, a power switch and a mains outlet socket. You can use extension socket blocks if you need more outlets.
The ratings - Your lights draw very little power but obviously, the battery will last longer if you use less. Any inverter rated more than your total load (all the Watts of all the light strings added together) will do. Most inverters are rated at 300W or more. A 300W one would power 12 of your normal light strings (288W total) or 46 of the LED strings (294W), that's an awful lot of lights!!
How long it will last - the exact figure will depend on many factors but you can make a rough calculation from the battery Amp-Hour rating which should be on it's specification. The rating means it can supply that many Amps for that many hours. For example, a 50AH battery could supply 50A for 1 hour or 25A for two hours or 1A for 50 hours. So what you need to know is how many Amps the inverter is going to draw, then you can estimate how long the battery will hold out.
The formula, at it's simplest is Watts/Volts so for example one LED chain would draw 6.4/12 = 0.53A and a 50AH battery would last 50/0.53 = 94 hours. That assumes everything is 100% efficient and in the real World it isn't so half that figure to be safe. You can still do a lot of romantic things in 47 hours
Aren't you supposed to use candles though ??
Brian.