Trevor Smith wrote:

give the phone kit enough juice to keep it powered up during
the time it takes to start the engine

I'd use 3 capacitors in parallel. 22uF (which is probably adequate),
then a 1uF and a 100nF all in parallel, because I'd be worried about
filtering high frequency electrical noise from the starter. Feed them
from a diode.

If 22uF isnt enough, then either fix the engine to start more quickly,
or parallel up another one. As we don't know just how much the
bluetooth gadget draws for current, it's not practical to calculate
this beforehand.