So I finally had a chance to complete the swap.
I took apart my i3 SP3, and unfortunately cracked the screen in the process. I used playing cards and a heat gun. I went very slow, but this had no effect. The screen cracked after a few centimeters of me sliding the card through the adhesive. The screen is more brittle than a premium plus cracker. In addition to this, the adhesive on these devices is insane. It's almost tar like and will not give at all. Luckily the screen was still able to display content which was helpful for testing.
Before i took apart the i5 surface, i made sure to update the FW to the latest version.
Once i had both the i5 and the i3 surfaces apart, i put the i5 board inside my i3 device. The i3 had the issue of "plugged in, not charging"
Unfortunately, the device still showed the same plugged in not charging error on boot. This is pointing to a battery failure instead of a FW issue.
The next thing i'm going to try is swapping the charging boards between devices. If this still does not work, then it would completely confirm a battery failure.
This is really unacceptable if it does end up being a battery failure, as my device only had 124 cycles on it based on the powercfg report.
Thank you BelalK this is very informative and helpful, particularly for myself. I'm not as adept at testing hardware/software but keen to take the initiative and try to work out the true issues. I have no idea which battery my Surface Pro3 had, however it was only only about 14 months old and had more often than not been directly plugged into the power supply at my desk. It died a sudden death (mid 2016) while travelling and relying on it's battery supply. It warned that it needed to be plugged into the power supply as the battery was low, which I did. However, it didn't appear to be recharging and was unable to switch back on. Obviously I lost everything on it and had no laptop for the rest of the trip. I then discovered the problem on the forum and was subjected to the same appalling treatment by MS and MS Store as others.
I'm keen to know the results when swapping the charging boards.