Somewhat educated guessing...
The alternator is probably at least 50 AMPs and possibly as high as 100A. A discharged battery will pull as much current as it can at first and is limited by the alternator capacity, wiring and internal battery resistance.
Most cars that I have checked, and it has been awhile, usually pulled between 5 and 10 AMPS while running. So that leaves some capacity for recharging your second battery. It may take longer to charge if for example your excess capacity is only about 20 AMPS(a whimpy example), but in general most storage cells only hold around 70 AMP hours anyway which would mean maybe 4 hours to charge back up at 20 AMPS. The charge rate actually drops as the battery gets more charge.
You probably have enough alternator capacity but will be limited by the size of wire you choose. I think you need at least something like 10 gauge to supply that current and since your capacity is probably higher, you want to size the wire for the maximum current you think the alternator will supply and the battery will pull. Fatter(lower numbers) is better when it comes to battery charging.
A couple words of warning. Those coolers pull a fairly heavy load, several amps at least. If you leave one running overnight, there is a good chance that your battery will be drained. Connecting batteries in parallel will help but leaves both batteries open to full discharge, say over a weekend.
You will need a battery isolator to keep the batteries separated and prevent your starting battery from getting drained. RV supply shops have these.
Another comment. I kind of like these coolers and own 4 of them. My experience is that they only work well in cool weather. When the weather gets hot enough to heat the inside of your car past say about 90 degrees(this is real easy to do) for a long time, then the cooler won't be cool enough. The coolers only cool to 40 or 50 below ambient. If ambient is 90 then your soda is going to be 50. A warm summer day(80 degrees?) will kick ambient way up into the 130 plus range. I think I even read that it can get near 200 if you believe the "honey I roasted the kids" news stories we get in the summer. No way for the cooler to keep up with that.
On trips across the desert with my AC on, these coolers will do a fair job if you like cool but not cold soda, but not nearly as good as an ice chest.
Just my opinions,
Jim Henderson