Solar Power Inside a Car
Firstly, solar panels inside a car is pretty pointless. Solar panels need to be in direct sunlight to be effective. Having them inside a car means they're not going to be in direct sunlight, so they're not going to be able to generate anywhere near enough power - unless they are massive, in which case where is the child going to sit? Yes, you could put them on the roof, but that would be silly.

The whole concept of solar panels for this project is completely wrong.

The car has a big battery and an in-built generator. Use them. There is absolutely no reason for you to have your own batteries as well. Yes, you could charge your own batteries from the car's battery, but that just introduces extra losses and wastefulness.

The only possible benefit of having your own batteries is it will safeguard the car's battery when the engine isn't running. It would then be impossible for your project to flatten the car's battery. But if you design your project properly that wouldn't be a problem anyway.

Some things you seem to be missing with your project:


The car has power you can use. It's throughout the car. No need to provide your own external power source.
The car has an "accessory" signal that is only turned on when the ignition is on. It's used by things like the radio etc so that when you turn the car's ignition off the radio turns off. No need for vibration sensors or anything to know if the car is running, it can tell you directly.
Cars generally have "seat occupied" sensors in them that couple with the seatbelts to know if a passenger hasn't got their seatbelt on and warn the driver (in most jurisdictions it's the responsibility of the driver to ensure all passengers have their seatbelts on).


If you really must charge batteries there are certain safety aspects you really must take into account. Your charge circuit must be able to manage the battery charging 100% safely. You have mentioned heat, and that is a very very important thing to manage. A decent battery manager chip will include an input for a thermistor. That is used as one of the multiple methods of charge termination (there's dv/dt (change in voltage over time), dT/dt (change in temperature over time), etc). It also serves as an over-heat safety cutout. Ensure that the charge circuit you use not only provides a thermistor input but also has the ability to cut off charging in the case of an overheat situation.

You also need to consider over-voltage. A car's 12V supply isn't 12V. It can get as high as 16V sometimes, and noise from the ignition can cause regular spikes as high as 70V or more. Automotive electronics have to be able to withstand those spikes, and that means your charge circuit has to be able to withstand those spikes.  Buffering the 12V through a proper 5V (cigarette lighter USB charger) regulator will remove that noise and make your circuit easier to work with.

So you need to think more carefully about just what it is you want to achieve and how that is going to fit into an automotive situation.