Maybe others with suitable experience will supersede what I say below.....
Finding the appropriate alarm output is a separate issue, but LEDs are typically 2 Volts each.
By saying you can cut into strips of 3, I presume you mean separate them into groups of 3 LEDs that have been connected in series with their resistors so that they can be powered from 12V....
IE - 3 LEDs = 3 x 2V - 6V, plus 1 to 3 resistors to drop the remaining 6V. 6V+6V=12V.
The actual strip connection method isn't too important as long as it is split into 12V sections. That might be 6 LEDs in series, or 5 LEDs & 1 resistor. It could even be 1 LED with a series resistor that drops 10V - but then 3 LEDs means 3 PARALLEL strings of LEDs.
But if yours is 3 LEDs as stated, so be it. And a single series "string" per door handle is assumed.
The wiring to each 12V LED string is straight forward - +12V to one end, and 0V (ground) to the other.
The +12V has to go to the Anode end (+), and gnd to the Cathode (K or -) end.
Getting these the wrong way around isn't destructive - it just won't light up.
Don't worry about voltage variation. Whether 10V or 16V for "12V of LEDs" won't matter much.
LEDs don't use much current - typically 10mA to 30mA per 12V string described.
The alarm output must tolerate the current required.
EG - 4 parallel strings of 10-30mA = 40mA to 120mA total.
You may wish to protect the alarm output with a fuse. This fuse should not exceed the alarm output's rating.
If the output is to control a relay, then a fuse should be used in the string feed in case of cable faults. Sharp metallic fembot & autobot fingernails have been known to pierce cable insulation and short to ground.
A single fuse can supply all strings. The fuse rating should be (say) 30% higher than the total current, but since its function is to protect the wiring to the LED strings, it can be rated up to the cable capacity.
For a 40-120mA total, I'd recommend a 250mA or 500mA fuse as these are common (but not as blade fuses?).
[FYI - The voltage-dropping resistor for each string could achieve the required protection if located at the strings' power source (alarm output or relay), but alas these are integral to the SMD tape array. But a low value high-power resistor could replace the abovementioned fuse.]
You'd probably want to switch the +12V supply to each string rather than have a ground-switched "hot" string. But that depends on the alarm outputs and preferred implementation.
You should test each string of LEDs across your 12V battery before fitting to test and confirm polarity.
I'd probably run each string cable from each door handle and back to a central point where your (optional?) fuse and relay else alarm contact reside - remembering that a relay can be placed anywhere.
The different strings can then be joined at this distribution point.
Likewise each string could be individually tested (and even polarity swapped) from the Distribution Point.
I hope that helps.....
A detailed diagram would require the controlling method to be known - eg - via a relay from a ground switched alarm?