The energy that the battery supplies is the integral of the \$V_{in}I(t)\$, which is just the total charge that left the battery during the charging time:
\$E_{battery} = V_{in} \cdot \int{I \cdot dt} = QV_{in}\$
All that charge got to the capacitor (since they share the same current), and the energy in the capacitor can also be written in terms of charge (using Q=CV):
\$E_{capacitor} = \frac{1}{2}CV_{in}^2\ = \frac{1}{2}QV_{in}\$
Subtracting the two will give the amount of energy lost to the resistance during charging:
\$E_{battery} - E_{capacitor} = E_{resistance} = \frac{1}{2}QV_{in}\$ which is the same as the energy in the capacitor.
The 10uF cap can be near the regulator where it does the most good. The inrush current with that size of capacitor should be no problem for a 3.5A fuse.
The fuse you have selected is a fast acting fuse but the thermal mass of the fuse will unlikely respond in the short time that it takes to charge a 10uF capacitor.
You will always have a certain amount of series resistance in the wiring, connectors, PCB traces that also will help to limit inrush current. Plus under normal circumstances I suspect that you would cycle the mains power switch of the 12V supply to power down your circuit instead of direct connecting the +12V. In this case the supply will have a fairly lengthy rise time at its output that limits the inrush current to same levels.
For grins I ran a simulation assuming a wiring resistance of 0.1 ohm and the 12V supply coming up to full voltage in 10us. Under these conditions the inrush current is ~12A for the 10usec rise time of the supply (linear rise used). My estimation that under such conditions the fuse material may over heat and blow only if the switching duty cycle was faster than the fuse can cool down from a 10usec pulse.
Do note that in the past I have had first hand experience of seeing fuses crystallize and fail after years of service being subjected to inrush currents. That was on the rectified DC lines of a Cromemco S100 chassis that had enormous capacitors. Your 10uF caps would look like specks in comparison. The Cromemco fuse in question was the 30A fuse on the 8V rail as shown (in the photo here). Inside the back of the unit the associated capacitor was the large soup can in the (closer part of this image). That capacitor was a 130,000uF / 15V unit.
Now if you had something like a 4700uF capacitor then there may be more concern. In that case you may want to select a time delay SloBlo type fuse.
In some electronics devices where there is indeed a very large inrush current possible the equipment is designed with a low ohms resistive device in series with the input. As the device comes up a special circuit either detects when the input caps are charged or just waits some nominal delay and then activates a relay that shorts out the low resistance device.
Best Answer
NTC thermister in the AC side of the Bridge rectifier is best .The big filter cap on the DC side gives some protection for the Bridge from voltage spikes.There is these days motivation to bypass the NTC when the system is up and running in order to save power .A relay is robust and simple and the coil wastes less power than the hot NTC .When the NTC is bypassed by the relay on the AC side the relay is switching AC where it will live up to its ratings .One NTC in the AC side is adequate .Using two NTCs one in each leg can be cost effective at high power and gives a more symmectrical circuit layout which can help EMC .