Cable
The cable you're looking for is Type UF, or Underground Feeder cable.
It can be purchased at big orange, and big blue, by the foot. It's available in 6/2, 6/2 with ground, 6/3, and 6/3 with ground, and is rated for direct burial.
Attachment to Service
There is one temporary, and two permanent ways to supply power to a park trailer.
Temporary
Power-Supply Cord
You'll want to use a NEMA 14-50 receptacle, and 4 wire cord with NEMA 14-50 plug to connect the trailer to the service. The trailer's electrical grounding will be through the service plug, so only the distribution panel will have to be grounded.
Permanent
Mast Weatherhead
You can use four continuously insulated, color-coded feeder conductors strung from a mast to a weatherhead as a permanent feeder.
Raceway
A metal raceway, rigid nonmetallic conduit, or liquidtight flexible nonmetallic conduit, from the disconnecting means to a junction box on the underside of the trailer can serve as a pathway to run permanently attached feeders.
NEC
For reference, park trailers are covered in article 552 of the National Electrical Code.
Is this approach correct?
Basically, yes.
1mm2 twin and earth cable (I presume that is what you are proposing) could carry up to 16A, depending on where it runs, so 1.5A is well within the headroom (assuming that you aren't talking such long runs that voltage drop becomes an issue).
For lighting, however, I tend to use 1.5mm2 in preference to cable 1mm2.
Best Answer
If you look more closely at the link, the bulbs are actually for mains voltages, in one of the pictures, it states "Input Voltage 90V-264V". This is the voltage that needs to be provided to the bulb, and the voltage is then internally stepped down to 12V for the LED modules. So in this case, the name of the product is misleading. If you look at the fitting, this is the "E27" fitting, which is an industry standard, and so you should expect any bulb that will fit in this fitting will be for mains voltage.
Answering your question about the cable, the cable is designed for use at 240V, and will have a certain rated current. If you stay within these limits, the cables should be fine to use at lower currents or voltages.
LEDs are normally powered by DC voltages. Either they accept a lower DC voltage, or have a built in circuit to rectify and reduce the input voltage. If you decide to buy LED lights that accept 12V input, a transformer alone will only give AC voltage. It is possible to find 12V DC plug in transformers which could work for powering LEDs, but you will need to check that it can provide enough current.
Also it is possible that you might encounter problems with the length of the cable. This will depend on the cable length, the cable core diameter, and amount of current you need to provide to the lights, but resistive losses could result in a loss of a few volts, which could mean the lights don't work, or at least won't illuminate fully.