What's likely happening here is when you switch on your 3.5A load, it is as first drawing much more than that. This is called inrush current and is often caused by capacitors being charged up to the supply voltage. Since you mention a car battery, it's also possible the initial current is high because you are starting a motor. A stopped motor provides no back-EMF and so will draw more current until it's up to speed.
This isn't a problem in itself, but all batteries have an internal resistance which limits the maximum current they can drive. Ohm's law states that as the current through a fixed resistance increases, the voltage across that resistor also increases. The practical consequence of this is that as the current draw on any battery increases, its voltage decreases.
Most likely, you see the current draw on your 100mA load decrease because the battery voltage has decreased so significantly that there is no longer sufficient voltage to drive 100mA through it. If your load were simply a resistor, you could use Ohm's law to know the relationship between voltage and current.
You could, in theory, solve this by putting a capacitor across the battery terminals, which could provide some stored energy to fill in transient demands such as this. Ostensibly, the capacitior would have a lower internal resistance than the battery, and it would temporarilly be able to drive the needed current, as long as it isn't needed for too long.
However, the internal resistance of a car battery is pretty small, and a capacitor large enough to store enough energy to supply this transient load is probably impractically large and expensive.
One solution is to limit the inrush current to your 3.5A load. Look for "soft" or "slow start" circuits as a starting point for your research. The other solution, if you only care about the 100mA load, is what Eric Gunnerson's answer suggests.
In an ideal world, where a capacitor has no series inductance and an inductor has no parallel capacitance, and voltage and current sources can provide voltages and currents with a step-shaped profile, the current into a capacitor and the voltage over an inductor can change abruptly.
Note that the reverse is not true: the voltage over a capacitor, and the current through an inductor, can not change abrubtly (unless you allow for non-finite currents or voltages, like a Dirac-shaped pulse).
Note that this ideal world is an mathematical abstraction, you can't buy such components.
Best Answer
This actually exists in RF transmitters like walkie-talkies. People call them "LC tanks." You're right that the amount of energy that can be stored is quite small. The reason people use them is to store energy that can be withdrawn as oscillations at a certain frequency.
You might Google "Colpitts oscillator" or "Hartley oscillator" for more information.