Finding a fuse with those characteristics is going to be difficult. I'd use a current shunt with a suitable MCU and circuit breaker and monitor the current taken by the device, shutting off the current if it rises above 120A for 15 seconds.
Two fuses in parallel can cause problems unless they are well-matched and the holders are properly designed. Any difference between them can cause one fuse to take more current than the other one, and fail.
The type and rating of the fuse you must use depends on your local regulations and laws, so it is difficult to help without knowing.
However, the regulations, no matter in which country, depend on physics - more precisely, they ask if the fuse will prevent the wires, switches and outlets from burning in case something fails and causes an overcurrent.
Despite your nice diagram, we can't tell what types (diameters/cross section) of wire are used in your installation, and where the wires are put. For example, different ratings apply depending on whether your wires are laid out in a stone wall or in a wooden construction, or in a wooden construction containing some thermal insulation making it harder for your wires to get rid of heat.
A typical installation in Europe, for example uses wires with a cross section of 1.5 mm2 and fuses with 16 A. If the cross section of the wires is smaller, the fuses have a lower rating, too. If the wires are put inside of thermal insulation material (e.g. in an outside wall or along the roof), you may have to use 13 A or even 10 A for the same 1.5 mm2 wire.
If you are not absolutely certain about your local regulations, please ask a professional electrician. If not done right, you put your house in danger of burning down.
Also, if you have sparks in your fuse holders, I think it's time to replace the entire fuse-and-holder combination. Sparks / arcing may cause very much heat, starting a fire in your fuse compartment!
Best Answer
The quick answer is that you should be OK, except that, if your 27 watt LEDs are 12 volt, they won't work, since in series each will only receive 6 volts. In comments Jasen suggested that the current drain may be too high. This discussion suggests that cigarette lighters are fused anywhere between 5 and 20 amps, so you'll need to check your owner's manual to find out just what your fuse really is. Also, I suggest getting rid of plug 2.3. The last thing you want to do is have someone plug in a lighter while the LEDs are on, since this will probably blow the fuse, and they are a pain to replace. Make sure any wiring is rated at the fuse current - Google on "ampacity" and use the smaller ratings for a given wire size.