3-5 volts is a very normal voltage drop of a service neutral between the meter and the transformer, this voltage drop will raise the voltage on all the grounding and neutrals of this building to remote earth, this pool most likely has some or all of the equal potential bonding missing, and either the water is at earth potential, or hand rail (verifiable by running a wire out to a point in the yard 30' should be enough, putting a short rod or screw driver into earth then measure the voltage from the wire to the water, or hand rail, the one that give you 3 volts has a bond to the EGC of the pump or the pump casing, now do the same thing at the house meter, measure the voltage to remote earth from the meter case or even the ground rod, you should see the same voltage, this is a very normal voltage drop and is almost impossible to eliminate as it would take the POCO to replace the service neutral with a very large neutral conductor or some other trick that in most cases they would be very reluctant to do.
So since fixing the neutral problem is too hard, then the NEC now requires that we short together all metal, concrete, and water so a difference of potential can not exist within the reach of a person in the water, reading article 680 can show some of the methods this can be accomplished, it's called equal potential bonding.
But for the most part this has to be done when the pool was installed, so now you have a bigger problem, how can you install this after the fact, not impossible but in some cases it would take removing allot of the concrete deck and exposing the metal shell of the pool or the re-bar in the concrete shell of the pool if there is any, then bonding it to all the metal parts and pump, heater, railings, and even the concrete deck around the pool by bonding the re-bar in it, but hopefully you can use one of my tricks below.
NO GROUND RODS at a pool!!!
This is one of the most misunderstood practices that many do, a ground rod will not remove this voltage that has many amps of current behind it and can not reduce this voltage, lets say you have a good rod and got the code minimum of 25 ohms, @ 3 volts will only cause 120ma of current flow, no where near enough to affect this voltage.
If this voltage goes away when the main is turned off then it is caused by the voltage drop of the service neutral, if it remains then it is a voltage drop from the primary neutral feeding the transformer, it could be a bad primary neutral connection (MGN) that the POCO can track down and fix, another method is to get the POCO to install a neutral isolator or neutral blocker, installed between the primary neutral and the secondary neutral, if this is a normal voltage drop on the MGN then it would remove this problem, but if it is coming from the secondary neutral voltage drop then the only way would be to over size the service neutral to the transformer.
here are some of the other dangers you are dealing with if you do not have an EPB around the pool:
a loss of a connection of the service neutral from the point of the main bonding jumper and the transformer will result in very high voltages between the water and this hand rail.
A large fault to the neutral or grounding from any hot will also cause the hand rail voltage to be increased.
the two above will cause the neutral to earth voltages at the service to rise much higher which through the EGC to the pool which could cause a high enough voltage to cause a person to be killed.
Here is my arm chair thoughts on what could be going on:
from what you have said, the railing is bonded to the pump which is bonded to the service neutral via the EGC ran to the pump, the water is at earth potential thus giving you the 3 volt drop of the service neutral??
my second version is that the pool installers did bond the pool casing to the pump but when the deck installers installed the deck they didn't install the bond from the pool or pump to the hand rail or the re-bar in the deck???? again then this puts the pool water at the voltage drop of the service neutral to earth and the hand rail along with the deck at the potential of remote earth???
running a small wire 30' away from the pool and connecting it to a small rod or even a screw driver stuck into the soil, then use the other end of the wire to see what is at the earth potential and what is at the potential of the service grounding (PUMP EGC), this will tell you what is bonded and what is not, then you could find a fix for what is not bonded to the service neutral putting a 1k ohm resistor between you leads will also keep you from reading stray voltage, I wrap it around the plugs right where it plugs into the meter, if measuring to the water the resistor will reduce the reading but shouldn't when reading to remote earth to the pump or hand rail, I would take readings with the resistor and without.
if it turns out that only the deck/handrail is the problem then talk them into a new deck this time with the proper bonding to the pump and pool, another fix and a cheaper choice is to have a concrete person saw cut a grove large enough for a #8 bare solid coper wire in the deck to the hand rail from the pump to lay a #8 solid copper wire in it with at least 20 feet of the wire to be in the grove, if this can be done in or at the location of an existing grove it would not be as noticeable, fill the grove back in with concrete and finish to match the existing, bond it to both the pump and the hand rail, this is only if the test above shows the 0 volts from the hand rail to remote earth but 3 volts from the hand rail to the pump grounding.
Let us know what you found