DC current through capacitor in Falstad
Looks like you found a convergence corner case.

Cap directly connected to voltage source: Falstad detects the cap loop and says no.
Cap connected via ground: Falstad doesn't detect the loop, oscillates (what you're seeing.)
Cap connected via resistor to ground: Varies. Below 100 mohm, won't settle down. Above 100 mohm, converges.

Here's my sim (clicky here):

Point being, avoid direct capacitor (and inductor) loops to fixed sources. These always cause trouble with SPICE and its derivatives.