Charging lead-acid battery in parallel with load - what if the system voltage goes above nominal, and how to prevent it?
Your system needs to cope with a battery voltage from about 5.5V through to 7V - these figures will depend on the exact type of battery. Short of adding buck or boost converters, there isn't really any way around this.

An Arduino is probably overkill, but you need some sort of undervoltage cutout.