I don't claim to know everything about batteries, but I am a recently-retired EASA Licensed (fixed wing) Aircraft Engineer. I'm not aware of a manufacturer-mandated in situ serviceability test, but I don't have Type Ratings on everything. The only way to definitively check the health of a battery is to remove it from the aircraft and perform a Capacity Check on the bench. This takes around 24 to 48 hours. If you're in any doubt about your battery's health, talk to your engineer/mechanic or your Continued Airworthiness Manager. A Capacity Check is part of all aircraft's ongoing Maintenance Schedule, but then, so is everything else that sometimes fails.
Simply checking the voltage of your battery either before or after engine start will not tell you its health. The massive current draw during engine start takes some time for the battery to recover, and the load-meter or battery-charging indicator willl corresponding show an initially very high current, falling off as the battery recovers; typically around a half-to-one-minute before the graph starts to level out. Your POH may include a minimum generator (engine) speed during this time to ensure adequate generator cooling, and should stipulate a maximum permissable charging rate (by implication, how much the battery has recovered from starting the first engine), before starting a second engine. Subsequent engine starts will further deplete your battery, and you may find that another limitation on battery charging current exists before take-off. Your POH will tell you how long you can rely on battery power after an airborne generator failure, but this assumes that the genny failed when you had a fully charged battery.
Another problem with a partially discharged battery is that engagement of electrical equipment with a high current draw may unexpectedly drop your bus-voltage below the minimum for some equipment. For instance, the DA42 had an issue where retracting the gear would cause the engine management system to drop off-line, leading to a double loss of engine power.