Electrical – Bulb replacement in a car 12V 60mA vs 14V 65mA

You should find the cause of the voltage drop as John U's comment suggests. You have a high resistance somewhere in the switched circuits (perhaps in a fusebox or on the battery connection) you should be able to trace this with your voltmeter with the hazard lights on.

A capacitor should go in parallel to the load (not in series), you'll need a high value (and therefore electrolytic or supercap type) rated for the sort of voltages that can be seen in automobiles (I'd look for caps specifically intended for automotive use, but I'd guess at 30V rating) The exact value depends on the current drawn by the total loads (unless you use a diode to prevent other loads also drawing current from your capacitor) the minimum voltage usable by that load and the time it needs to be sustained. See RC time constant. Temperature rating depends on location in car and your local climate (As I'm sure you know, a car left in the summer sun all day in Arizona will probably get pretty hot inside, maybe very hot under the bonnet)

With automotive electronics you need to think about what might happen if the battery is connected backwards or a 24 V battery being accidentally connected. You might think this will never happen but one day you might accept a jump-start from a stranger late at night in the rain.

It is unlikely that you will damage the electronic module because your bulbs draw less current than the proper bulbs. However, they will not last as long as bulbs of the proper voltage rating.

The problem is that the DC voltage in a 12V automobile sits right around 13.8 Vdc when the engine is running. Bulb lifetime varies by about the 12th power of the change - that means that a small change in voltage makes a huge difference in bulb life.