Power is Current * Voltage.
Your saw needs 13 Amps at either 110V or 230V,
This means the power (using 110V) is
13 * 110 = 1430W
The supply to the inverter has to supply the same power, plus the losses - assume 10% just for illustration, so now:
1430 *1.1 = 1573W input, at 12V this means
1573 / 12 = 131A
1) The inverter is too small, especially as the starting current is usually higher
2) A single battery will do this, but it has to be a large battery
3) I would not advise having the engine running - a large load like this could damage the alternator
I would suggest a second battery or batteries...
Or, a genset...
What you describe is a manual split charge version with a switch. There are automatic systems that will connect to charge the second battery once the engine has started - which means you don’t forget.
The split charge systems can be found on the bay of fleas or places that do caravans are also likely to have the parts, so do places that suppy and fit winches. A competant vehicle electrician can fit this easily.
Also see this answer and question : https://mechanics.stackexchange.com/a/53030/10976