Arduino shows voltage drop, voltmeter doesn't
According to your schematic, when you close the switch you will short the +5V 3A rail to the 2.6V battery rail - this is not good unless you are controlling the charge current (Have you measured the current?)  

Your wiring/switch could account for the drops between inputs if there is a lot of current flowing - for example at 3A, you only need 3A * ~330m&Omega; to give you a drop of 1V.  

More info would help, e.g. on the type/capacity of batteries, type of supply, how you are confirming the A1 A2 voltages, etc.
A constant current charge is preferable. It's also possible your bench supply is just showing you the "set" voltage, but if it's in constant current mode the actual voltage will drop as the circuit draws more current - I would measure voltages with a multimeter at various nodes to make sure of what is happening, and also measure resistance across the switch to the supply (i.e. between the A1 and A2 nodes)