L;
Great! The new inverter is working better. I believe the main reason is that it has a lower Voltage shutoff of 10.5. This is a guess because your 600W was OK with the 580W cooker, except when at idle when the Volts dropped too much.
At this point, you can continue on with what you have now, so long as you know you must drive, or fast idle, when cooking.
Now you need to get a plan. I suggest:
1. Get the alternator and accept MEX's offer of help.
2. Shop the new battery and be ready to replace the old one. Look at
DEKA to get an idea of what specs are available. I used the 8a24m for 6+years as a deep cycle house battery. A friend installed the 8a31dtm in his Tundra(tight squeeze) so he could run a 12V fridge in the bed.
What is your tray size? Can you fit a 13"+ battery like the 8a31dtm?
3. Once you get the AGM, run some tests with the cooker and tools and with engine off. If/when that fails it will be because of inadequate wiring. :)
4. There are better choices for solar. Even Home Despot has some $160 100A kits. I had two 85W panels on the Safari that gave up to 9A. After it was wrecked, I moved them to the travel trailer.