Turning led strip on with car battery power
I've found out it can be done with a resistor but it's not the best way to do it, since it will get dimmer/brighter depending on the voltage of battery at the time.

Yes. If they're 12 V strips then this would be a bit of a problem.

Also found out it can be done with constant-current led drivers (made with LM317), but the problem is that these regulators have a voltage drop (3v for lm317 I think).

The CC driver isn't appropriate as the strips are designed to work at constant voltage - 12 V in this case. If you were to use the LM317 you'd use it in a voltage regulator configuration but, as you say, the voltage drop is too much.
The LM317 is a forty year-old design and things have moved on.
Have a look for a low drop-out (LDO) voltage regulator with a 12 V output that will handle the current you require. These are optimised to minimise the drop-out when the the input voltage is low but otherwise work like a regular voltage regulator.