How to power Arduino and shields with 4 rechargeable AA batteries?
Most 5V devices will function fine at up to 5.5V - however, check the datasheets for the major components and modules on the boards.

The main thing to watch though is that a 3.3V device, which can normally communicate fine with a 5V one, may not be able to. The problem is that the logic HIGH signal from the 3.3V device may be below the threshold for a logic HIGH on the Arduino:

V_IH = 0.6Vcc = 0.6 x 5 = 3V

V_IH = 0.6Vcc = 0.6 x 5.5 = 3.3V


Thus a 3.3V device would have to output at the very minimum 3.3V as a logic HIGH in order for it to register as one on the Arduino, which may be a problem...

You can easily drop the battery voltage by 0.7V though by inserting a single silicon diode (such as the trusty 1N4001) in series with the batteries. That will take your voltage down from your measured 5.3V down to 4.6V which will improve matters somewhat.

However, it is important to note that you have so far measured the open circuit voltage of the batteries. That is, the voltage with no load on them. The voltage will most likely be lower once they are actively driving the Arduino etc.

The Arduino itself will be fine running off 5.3V, so I suggest you power just the Arduino direct into the 5V pin from the batteries and measure the actual voltage while powering the Arduino. That will give you a more accurate voltage reading. Then, if needed, you can add the diode to drop it by 0.7V.