I agree with enblethen, you have wiring inadequacies more than anything. It may be something as simple as a poor ground connection at the battery, and cleaning it up would help things. Connecting your inverter directly to the battery would also work well, but be inconvenient. A more permanent solution with direct wire connections would be ideal.
Your meters are reading voltage, but not necessarily the voltage of the battery (this is particularly true for the one with the tank level meters). It's quite possible that they're also seeing the voltage drop from the inverter's current consumption. The battery voltage itself is probably rather higher than it suggests while under load.
You might also want to check the condition of your batteries if you have not done so yet.
