So would high current through a capacitor cause heating?
Yes and no.
Capacitors only allow current through them for a very short time - until the plate charges up. At DC no current will pass, no heating.
At higher frequencies (eg 40kHz) the inductance of the wires will prevent much current from being available so again, you get very little heating.
Every capacitor will have a low point, a charge/discharge frequency where heating is maximized. Even so, any reasonable capacitor will survive so long as there's some ventilation - they're designed that way.
If you want to minimize the heating at a certain frequency you can mix different capacitor types, put multiple capacitors in parallel, etc. It's not difficult to fix.
But ... before you try to 'fix' it you have to find out if it's actually a real problem or not.
It's a complex situation (the inductances around the system are unknowns) so the best way to find out is to build one and see what happens!