You should get full battery 1 voltage at the purple/w stripe. What is the battery 1 voltage right now?
If you applied 12V to the purple/w stripe wire and have a good ground you should here the relay energize and since you have 24V on the red wire if the relay energizes you should have 24V on the purple wire.
Your relay sounds like it is bad if a full 12V did not actuate it.
Relay coils are designed to operate at a certain voltage. When the coil is energized at a the correct voltage there is a rush of current as the magnetic field builds. Once the field is built it limits the current flow through the coil. The when you remove the voltage there is rush of reverse current as the magnetic field collapses. There is also a drop out voltage where the coil will 'drop out' if the coil voltage decrease below this value.
If you have been sending 10.46V to a new, good coil you are not going to get the full magnetic field which can either prevent the power contacts from closing or not pull them in firmly causing arcing on the power contacts which destroys the relay's ability to switch the high power side.
If you have poor connections leading from the 13.2V battery 1 voltage to the purple/w stripe wire on the relay coil then that 10.46V many not be high enough to pull in the relay or it may pull it in but not hold the contacts tight leading to relay damage.
What relay are you using (manufacturer and model number), perhaps we can find the spec's on it to see what the coil pull in and drop out voltages are.
Your batteries are both fully charged right? You cannot troubleshoot electrical issues with batteries that are not fully charged.
This site has a nice description of a relay.
Relays