With no load, your tank battery supplies the 12.5v. Once you fire up the inverter and computer, the tank battery is unable to supply the required current and the voltage drops until the diode is forward biased and turns on. At that point main battery is supplying the bulk of the current but the voltage is now too low to make your inverter happy.
Since you're running an athlon and deskto HD, your system is probably pulling around 100W. Add in an inefficient inverter and your total power demand at the battery might be 150W or more. That's more than enough to make the voltage on your tank battery drop.
Then again, I have no idea what I'm talking about.