You need to look at battery and first determine CCA on it and what's OEM in the car if not original motor then what comes with that motor. Bigger number is better. If the battery has lost the CCA number they commonly use like 500 but that could pass a battery that is not correct for the car. I commonly look up in the app manual when no number is present or it is suspect, be advised most counter people will not lift a finger to do that. The battery life if 3 year 4 year 7 year ,etc, is a giveaway there too if battery has that still.
I always test battery AND alt, the tool used in the same and the battery could show as good if alt just spit guts out in the last five minutes. Rare but possible. Alt and regulator are inseparable, the alt could easily be bad and now going to toast new regulator with it, happens all the time. A dying diode or more pushes volts/amps up on regulator which is trying to make up for it, regulator then fries as well.
Get battery tested again, sometimes they pass and second time they don't especially if it's gone a bit downhill.
If battery on 4th year you aren't losing much cash changing anyway unless you have warranty paperwork, sounds like no here. In what I saw, battery life is the luck of the draw, I had a Walmart 3 year lowest of the low battery last for 11 years, you cannot run numbers any further really than at 4 years they are getting old, many, MANY fail around there. Maybe 60%-70% If you leave both cables on battery all the time when they sit then whack another year off that. 3 years? Maybe 50% OF that number I just gave, sliding ramp there, the numbers do not equal 100%. Very few make the 7 or 8 years mentioned, I'd say less than 10% That is modded by fact that everyone has second car now to not lose the all important job, if it sits long periods you are flat killing battery if not disconnecting at least one cable or trickle charging. I did it and watched battery life on the sitters go up like 50%. Cars that sit especially if a PCM on them can commonly toast battery in under 3 years, I saw it hundreds of times. Ask if they disconnect and the answer always no. One of the reasons why batteries took a whopping like 30%-40% jump several years back, they will tell you it was EPA lead disposal issues but actually it was all the goofy people who think you can leave a brand new battery in a collector car for 6 months and 'it should crank up because battery is brand new'. I watched hundreds of batteries warranty at 2 year or even 1, they just get all in your face about it and 'ka-ching!' let the cost go up.
Well, now I see it looks like alt WAS checked, the reg number 16.5 is it......good deal. That points more to battery as the problem as well. You need to check alt/reg again, it should drop to no more than 15. If old school alt 13.5 maybe as the low number. Those don't charge as much as they do now. Uh, anyone check water in battery? At 16.5 volts overcharging it would have been boiling off. Could now have a dry cell.......... .......
I had people who come in and refuse to let you look up battery for them, rather trying to find the group used and then pick the lowest CCA they could find because it was cheaper and 'I have no money'. It got to be a huge problem, the new battery will actually crank a bit more than that but they all settle in in like a couple of months to drop that cranking edge then they come back trying to warranty for a new one claiming its gone bad. It then passes the load test based on the CCA and they go to critical mass screaming and claiming that some other employee there sold them it as the 'correct' battery. We issued enough free upgrades to bigger battery they didn't pay for I lobbied the district manager to have us look up every battery sold and to refuse any one they simply threw up on the counter. That stopped most of the loss but they sure got mad about it. So don't buy cheap battery you have fooled yourself 80% of the time.
Don't discount cable interface, the cables corrode inside sheath to go bad gradually. You have to ohm them out with them separated on both ends. If bolt on ends the two bolt connecter used can corrode cable on the bottom to affect things too. As well as one inch away terminal to post. All must be 5 ohm or less resistance. I use no corrosion protectors at all, rather like wheel bearing grease to coat the interface, i get it underneath too. Messy? Wipe it off. I've seen weird atmospherics and maybe screwy terminal lead to make an uncoated terminal corrode to where you cannot see it at all but car does not start after sitting 24 hours with new battery in it. Just weird. Use grease and problem is gone forever. There are certain atmospheric conditions that mess up bare lead connections really quick depending I think on the alloy mix used there.
Check battery itself while just sitting, you have a voltmeter right? Any battery at 12.3 volt or less simply sitting is close to, if not exhibiting a problem. New battery at proper room temp is 12.86 volts, they work well until around 12.3 (after charging and one hour wait to discharge the surface charge all alts put in there), where some show trouble where some don't. I've seen some screwy setups like certain Hondas crank reliably at as low as 11.8 but not the norm. 12.3 is a number to note if you don't like to walk. I used it on hundreds of cars and the number holds up well.
The starter can get to where it draws too much amp too even though it seems to work right, that can make the problem too. That test done with the same tester used on battery as well if you hit the big chains. My harddrive getting cloudy, thinking more than 150 amp is bad IIRC.