This advice is applicable for those who do not want the design challenges or risks of working with lithium batteries or sourcing reliable protection circuits. There are simpler ways to go about this because a lead-acid charging system is already present; i.e. the alternator output is just right to charge a lead-acid and close enough to charge NiCd/NiMH.
Auxiliary battery (which can be portable)
Choose a battery which leverages the fact that your charging system is just right for a lead-acid battery. The obvious choice is lead-acid, and they come quite small. However, as it works out, ten NiCd or NiMH also will charge quite effectively off a system intended for lead-acid charging.
Of course, this should not be permitted to drain your main battery when the car is off. Nor allow the little battery to "help start the engine" as that would cause excessive overcurrent. Fortunately the annoying misfeature which cuts off your child's movies is well suited to this: hot-in-run circuits cut power in "off" and "engine start". Watch out if moving this to a car that doesn't have that -ummm-feature.
The only other issue is charging overcurrent if the battery is depleted, but that will vary by battery. A well chosen resistor will take care of this.
You could add a Schottky diode if you were particularly worried about reverse current.
Now, let's talk chemistries.
6-cell lead-acids are readily available in packages under a pound. The upside is it's a perfect electrical match and will perform well, and are cheap! The downside, besides their notorious weight-to-capacity, is they really do not like to be bottomed compared to other types. Unlike Lithiums where 2000 25% cycles and 1000 50% cycles are equivalent, lead-acids are much more aged the deeper the cycle goes. I am already talking about deep-cycle types. Don't believe the salesmen (unless they'll give you a warranty), talk to the solar, narrowboat, vanlife people: "size for 25% drain, 50% rarely" is common knowledge. Refs: a b c d e They are also damaged by being left in a discharged state - and can freeze if discharged.
It's a happy accident that 12V lead-acid charging systems are "close enough" for a 10-cell nickel-cadmium or NiMH pack. They will not overcharge and will slightly undercharge, but that doesn't present a problem. I use common 12V solar charge controllers to charge a large NiCd wet cell that is almost 40 years old, works like a champ. And like most battery chemistries, NiCd/NiMH aren't hurt by bottoming, in fact factory advice is to store them discharged. (that'll happen anyway, they self-discharge over 3-6 months.) Both NiCd and NiMH can output a reasonable amount of current if asked. NiCd capacity is better than lead-acid but not fantastic; NiMH is better. (keep in mind you will not be finish charging these to absolute max). None need charge or discharge protection circuits.
Lastly, since portability is a goal, I'd want to fab this into a package that attaches to the DVD player. (and maybe houses a couple of DVDs). Packaging is half the fun! I'd rather try to site ten C-cells than an SLA, honestly.
Time-delay relay to an always-hot connection (NOT portable)
This assumes you can alter the vehicle. It is likely they provide grab points to get the necessary power, securing the new wires is up to you. In this case, you use the main battery, but add an accessory outlet (or alter the existing outlet) so it is fed by an "always-hot" circuit through the relay contacts.
The relay coil is tied to any "hot-in-run" circuit, energized when the engine is in "run" or accessory (not off or cranking). That closes the relay, and holds it; when power is removed the relay stays closed for X more minutes, then shuts off. You need to find a value of X that makes sense for you.
The purpose of this is to avoid running down the main battery from accessories left on all night, so you can afford to be fairly generous - 30 minutes would be safe for a DVD player.