Will a trickle charge lead-acid battery system last?
As Mrs Beeton would say - first catch your battery. The sealed lead acid type used in a myriad of long term applications such as UPS and alarms. Generally this battery is designed for exactly this type of application and will maintain its charge losing something in the region of 40% over a period of 1 year. (Some do better, some do worse - check manufacturer's specs. It also depends on other factors such as temperature etc. etc.) You should expect a few years of service before changing the battery. As your running a wifi system why not send data about battery condition.

The trickle charge only needs to keep the battery topped up. Overcharging is always a potential problem so the trickle charger should have limited voltage.

The circuit you have shown would not work as it does not switch between the normal supply and the battery backup. You need something like this. Reg 1 limits the charge voltage and R(trickle) the charge current. The LED indicates the supply is on (and ok) Reg 2 is the normal regulated supply controller. The diodes allow the battery to take over if the supply fails. (switchover)