Works with Magsafe, shuts down on battery (96% charged)

Would love any tips on troubleshooting this situation. Macbook Pro 15" Retina Mid 2015 - A1398_820-00138, has battery A1618. It works fine with Magsafe connector plugged in. Battery shows healthy with 95% charge(checked with coconut battery too). However it powers-off the moment magsafe is removed - i.e. will not run off of battery.

I have Electronics Engineering background and am quiet comfortable with micro-electronics troubleshooting. Have repaired quiet a few appliances/gadgets/computers etc. So I decided to troubleshoot this myself so I can learn something new. Found Louis's youtube channel and it was great to inspire some confidence into me... I bought the needed tools - cheaper versions of what Louis recommends as I wanted to just do it for myself(not work as pro) for learning. I got a USB microscope, tweezers, bought the schematics and board-view. I had multilmeters, old soldering iron, but may upgrade it to a better one for SMD type work, and may get a cheap rework hot air-gun if needed. I built a DIY arduino oscilloscope as didn't want to plunge more money into this. I understood the charging circuit and all the SMBus communication stuff.

Here is what I have done so far :

1. First did the SMC reset, PRAM reset etc. with old battery.

2. Old battery - cell voltages with no power applied. cell-pair#1: 12v, cell-pair#2: 8v, cell-pair#3: 0v. At Battery Connector J5 Terminal 1-4, the voltage is about 2V.

3. With this old battery, when the magsafe power is plugged in, I can see the charger putting out ~12v on PPVBAT_G3H_CONN at Battery Connector J5 Terminal 1-4. The Battery indicator light is green. Battery shows ~96% charge, and shows as "Normal" in the Energy Saver Panel (same in coconut battery app)

When I unplug the magsafe, PPVBAT_G3H_CONN at Battery Connector J5 Terminal 1-4 drops down to about 2v

4. I got a new OEM battery from eBay(reputed seller) which incidentally had very similar cell voltages as my old battery -i.e. cell-pair#1: 12v, cell-pair#2: 8v, cell-pair#3: 0v. Also at Battery Connector J5 Terminal 1-4, the voltage is about 2V.

5. When this new battery was plugged in, the charger is putting out ~12v on PPVBAT_G3H_CONN at Battery Connector J5 Terminal 1-4. The Battery indicator light however is amber. Battery shows as 46% charged and displays warning - "Replace battery soon" in the Energy Saver Panel(same in coconut battery app)

6. I have checked voltages on R7150 and R7120, but found they were fine. Note that there is no board view for A1398_820-00138, so had to use the next one close to it which is A1398_820-3787. It was bit of learning, but could identify most of the test points.

7. Again did SMC reset, PRAM reset etc. with new battery - no change in status reported.

8. It is quiet possible that the new battery was faulty - I was able to return for a full refund.

9. I researched battery repair and understood the SMC world a bit more (http://be2works.com/hardware/) . The Battery SMC Chip is - TI BQ20Z451. I was about to plunge into getting the adapter (costs about 30) to do SMC chip reset, but held back as it was not clear if I had to unsolder this chip or it could be done via the battery connector. Plus I am constrained by time I have on hand.

However I did understand how the battery and SMC is supposed to work - i.e. battery health is manipulated by SMC chip and that is what Macbook will see.

Any pointers to where I can look next ?

1. Is it the battery or the board ? Anything could be wrong at board level that I can check ?

2. The battery A1618 for this model is not cheap - runs about 80-130$. I am trying to get a working battery A1618 locally(one guys on CL has it, but will not get it till after next week) so I can test it and buy only if it works. There is another battery A1517 for the A1398 models of 2012-2014 which has same power specs and sells for ~40-60$. However its battery connector is located at about 15 mm offset from center compared to A1618 battery connector I have - possible to do soldering hack, but not sure if its worth it.