Arduino PSU with battery backup (SLA) and mains/battery monitoring
Don't even attempt to build your own battery charger. Battery charging is a very complex problem with expensive stakes, and you should not do it as an afterthought to a design that's really about something else.   

Buy a COTS 3-stage battery charger of good quality, because lead acid batteries are quite sensitive to mismanagement.   Your scheme will require constantly adding water to the battery, and on an SLA that's pretty hard to do.  

The COTS charger also has a UL listing, which dramatically eases getting your own listing, since you are not dealing with mains power. 

Place your circuit in parallel with the battery terminals, so it is always powered, usually by the charger.  Focus on conditioning that 11-15V battery+charger voltage for the Arduino.   

Also think about load-shed as battery voltage falls.  Using more than 30% of a lead acid battery's capacity is damaging to the battery if you do it on a regular basis.  You don't want that.