Battery cables at the battery.
If either cable grew white or green corrosion in the past, simply cleaning off that corrosion does not make everything fine and dandy. Next time it acts up try torquing the stranded wire entry into the post clamps, you might find it suddenly is charging normally again.
If so, consider it temporary.
The only new part you should throw at it is new battery cables, and beware of the autoparts store prepackaged junk. SAE gauge wiring is 6 to 12% thinner than AWG. AP store battery cable ring terminals used stamped steel with a hole drilled through it. If you want top quality custom cables made:http://www.genuinedealz.com/custom-cables.
Their prices are good considering the top quality parts and tools used to make them.
Fusible link is supposed to look burnt and stretch out when it blows. I've never seen it actually do this. I have seen it look perfect and be highly intermittent or outright open circuit. Higher loads will cause it to go intermittent and lesser loads will have it function 'fine'.
An extra battery will allow you to drive longer with no functioning charging system, and will help to maintain higher voltages should your loads exceed that which the alternator is providing at that time. Depending on the size of the extra battery it is also another 50+ Lbs to accelerate and stop. I cycle the piss out of my single battery using a lot of electricity each night overnight, I have room for 2 more group 31 batteries for a shitton of total battery capacity and cranking amps, but don't bother. I'd rather have more charging amperage and manual voltage control. Voltage is electrical pressure, higher voltage more amps can flow into a depleted battery, or a not so depleted battery. Most voltage regulators are not concerned with recharging fully promptly, they are designed to never overcharge. The VR controls alternator output.
I don't know how much juice(amperage) is used to keep an AC clutch engaged.
but headlamps are about 15 amps, high beams are an additional 3 to 4.
The Hvac blower motor on high can be 15 to 20 amps.
At hot idle speed with these turned on, what voltage can it maintain? Does revving the engine bring it back above 14?
If the AC alone is dragging voltage to the low 13s and it keeps falling, does turning it off allow 14's?
When is the last time you put your battery on a 6+ amp charger overnight, and hopefully longer?
Many people assume a short drive is all that is needed to top up a battery, but an 80% charged battery cannot be brought to 100% charged in less than 3.5 hours, and those 3.5 hours assume a battery being held in the mid 14 volt range. Mid 13 volts will take 2 to 3 to 15x as long, depending on the health of the battery. listening to a stereo for a short while can deplete a 100% battery to 80% in minutes, depending on the stereo and volume, yet returning the battery to 100% cannot be done in less than 3.5 hours under ideal recharging conditions, which never occur underhood.
All lead acid batteries are happiest when kept cool and fully charged, hot and undercharged reduces their capacity and performance, and the hotter they are on average, and the lower their average state of charge, the faster they degrade.
Cold weather does not kill batteries( unless they are so discharged they can freeze), cold weather reveals weak batteries as the battery has less capacity and cranking amps when cold, and the engine requires more CCA to turn it in cold weather. A fully charged battery will not freeze until the -40f range or lower, but the less charged it is the higher temp it will freeze at.
The voltage one sees when probing the battery terminals after starting the engine is not the voltage the voltage regulator will always allow. Once things heat up it might never exceed 13.8v. One should check their voltage more regularly and at different engine temperatures/ rpm and speed. lots of underhood airflow at highway speed will keep an alternator much cooler than sub 25mph will.
My alternator will not exceed 135f at 65mph when maxed out at 1950rpm, but maxed out idling and it quickly climbs to 220f. Exceeding 220f is very hard on an alternator. Idling to recharge a depleted battery which can accept high amperage, is not recommended. Always plug in to fully charge when the battery is known to be weak or needed a jumpstart, and idling, just to recharge, is never wise.