I would doubt you have this type of charger, but it's worth pointing out that some automotive type chargers (high end microprocessor controlled 3 stage chargers and, with an equalization phase) for flooded wet cells can briefly take the voltage up to as high at 16.2 volts..
Since the microprocessor knows exactly what the charge VI curve looks like over time, it can take it up to this high a level to do a fairly quick and safe equalization on a seriously run down battery and typically for only a brief e.g. 15 minute period before ramping down..
Obviously these chargers were never meant to be doing this while sensitive electronics are hooked up to the battery..
The important thing is that you know what your charger max voltage is during during bulk, absorption and float in relation to the max voltage the regulators (that I'm assuming) are on your cicuit board can take.
Second, you need a charger built for driving loads in addition to the battery, otherwise (inexpensive) chargers can hang in stage 2 absorption mode and cook the battery..
It typically takes a microprocessor controlled charger to not hang in stage two and go on into stage 3 float mode.
Here is an example of a charger that would work fine in your application that I have some personal experience with..
Here is a simple tutorial I put together for for my ham radio club. If you need to know more about battery chargers in general, this might helpful w/o getting too deep..
btw you values for 1, 5 & 6 sound low really low..
Either you have an inaccurate meter or the battery is in bad condition. Are you using a known accuracy digital volt meter to measure pure DC & DC plus AC rms values?