I just measued by own 7 Ah gel cell. This is what I measured using a Hanger 9 charger (nominally 12V/600mA):
battery voltage no load: 13.4 V (it is fully charged)
charge voltage when the Hanger 9 charger is attached: 15.2 V
charge current: 300 mA
(p.s those number don't make complete sense to me...they say that batteries internal resistance is 6 ohms. higher than I would have guessed?)
So... If I connected this to 13.8 volts the current would be lower.
Repeating the excercise with a discharged battery would be more informative. (But I don't have any easy way to discharge my battery at the moment.).
If you do try the direct connect to your car, and the fuse blows, a resistor in series would limit the current. But if you're uncomfortable with it, don't sweat it. Just get the inverter.
A battery at a voltage potential of zero is practically a dead short circuit and tons of current will flow through it. I hope you realize that the power produced will be substantial and the battery will get hot instantly and could lead to serious injuries. So even if you had a 13.4 charge voltage you will STILL need a way to limit current. Also it's not very wise to discharge a lead acid battery completely as this will ruin it.