Do all Solid State Relays function as diodes?
Some SSRs do work like that, some don't.  If the output leads are marked with a polarity, then it matters.  There are "AC" SSRs that are two back to back light-activated SCRs.  In that case the polarity doesn't matter.

One issue with SSRs is that they can have significant voltage drops, from a few 100 mV to over a volt.  You don't want that between the charger and the battery anyway.  Even if this were a mechanical relay, you still wouldn't want it between the charger and the battery.  You'd then only be able to charge when the unit is on, which makes little sense.