Slow start up of LM3578
I would focus on the coil. How did you select 390 &micro;H, and what is its current rating?

Your symptoms are actually a bit contradictory. The excessive sag in battery voltage suggests that the circuit is drawing far more current than it should during startup, while the slow rise in output voltage suggests that the current is too low for some reason.

If your 390 &micro;H coil is going into saturation, that could explain the excessive current draw, while still being consistent with a slow start-up.

But based on a quick back-of-the-envelope calculation, 390 &micro;H seems to be too high for this application, by about an order of magnitude. Your 1 nF capacitor sets the oscillator frequency at about 80 kHz, which is a 12.5 &micro;s period. The maximum duty cycle of the switch is 90%, or 11.25 &micro;s.

With a minimum input voltage of 3.5 V, an output voltage of 12.0 V and a load current of 120 mA, the coil current is going to need to peak at more than

$$2 \frac{V_{IN}}{V_{OUT}}I_{OUT} = 2 \frac{12.0}{3.5} 120 mA = 823 mA$$

Note that this is already greater than the 750 mA that the chip is specified to handle. You're overloading it, and this is going to cut into your overall efficiency, dropping as much as 0.9V across the switching transistor.

But, in any case, this means that the inductance needs to be less than

$$V_{IN} \frac{\Delta t}{\Delta I} = 3.5 V \frac{11.25 \mu s}{823 mA} = 47.8 \mu H$$

Your 390 &micro;H inductor is acting as a current limiter, severely restricting the amount of current that's available to charge the output capacitor.