I don't believe that the power is totally cut off when the ignition is started.
When the key is turned to the "start" position, many "high current" circuits are disconnected. Such as wipers, lights, rear defrosters etc. to avoid robbing the starter from the all important start up. Easy to see by activating wipers and starting the car. The wipes will stop mid flight until the key is released to the run position
If the GPS is reseting, then it is most likely connected through one of these cut offed circuits. The OP needs to find a circuit that is "hot at all times". If the direct to batt is used then an inline fuse must be incorporated. On board modules and memories like radios etc are kept alive by using "hot at all times" circuits. The necessary currents are typically in the low tens of milliamp area, however, I have seen TSB's for some high end cars have been known to bleed down a battery if stored over long periods of time.
Reserve capacities of car batts today is very limited, headlights for maybe 10 minutes as apposed to the old days of 45 minutes +. To save a buck, they only design in what they need.
What normally happens is, the battery voltage drops as the huge current is drawn.
This is true...
However, a good battery should not drop below 10 volts, if it does then it is discharged or toast.
On a cold day, a starter could drop it down closer to 9 volts. Around 9 volts, the starter will just groan. Why block heaters save/extend a batterys life. Remember that the battery is only used to start the car. A working alternator keeps it going.
A lot of automotive "add on" devices account for this by regulating the 12 volt batt to an internal working voltage that accommodates the potential drops in voltage during these peak surges.
cheers