Label Vc and Vl in your diagram... current always flows from higher potential to lower potential (voltage). So depending on the direction your draw your current flowing, the voltage always drops in the direction of that current flow.
Simply put, in your first picture there is only one current loop. The current flowing through all the elements must be equal. I'll call the upper left corner node Vc and the upper right corner node Vl. Then namely:
(Vc - 0) / Zc = (Vl - Vc) / R = (0 - Vl) / Zl
Here Zc and Zl are the complex impedance of the capacitor and inductor respectively.
Alternatively, if we assume a current i0 is flowing through the circuit clockwise, and remember that voltage always drops across an element when current flows through it, then:
Vc = 0 - i0 * Zc
Vl = Vc - i0 * R
0 = Vl - i0 * Zl
Therefore:
0 = Vc - i0 * R - i0 * Zl = - i0 * Zc - i0 * R - i0 * Zl
Changing the sign:
i0 * Zc + i0 * R + i0 * Zl = 0
So yes, all elements should have the + sign.
You might want to use an inductor after all. Besides limiting the inrush current, it will improve your circuit's power factor which right now should be reduced by the capacitive nature of the load, at least judging by your description.
Of course, you'll need to find out how much of capacitance you have to counter to calculate the right inductor value. You'll have to either obtain that information from the datasheet or measure it. If you can't, just pick a value and try it out (I'd expect something in units or tens of mH to work).
Best Answer
It does make sense in the way that both the relay and capacitor would work fine in that set-up. There would be a large current spike when the capacitor is empty and gets connected to the supply, but it shouldn't last long enough to melt the wires. It will, however, cause the supply voltage to dip down, and that could mess with other things if they rely on it (like the Arduino could reset).
But I don't see any reason in your description to use the relay instead of a transistor to control charging, so in that way it doesn't make sense.