Sorry to hear about your troubles. I feel your pain. I went through the same thing about two years ago. I have a SP3 with a Simplo battery, too.
My SP3 works now, and I did not need any hardware changes, although, as MS was figuring it out, they also offered me the $450 (+/-) refurb unit for mine. I didn't take it. . .
Here is what I learned . . .
1) It was a driver problem, not a battery problem. In layman's terms, during a routine update, my surface 'forgot' what a full charge was, and would stop charging the battery when it reached a very low percentage. As I recall, that was less than 10%.
2) The power supply that the SP3 comes with is NOT strong enough for the surface, when the surface is running at or near full capacity. During those times, the surface will draw from the battery to compensate, then recharges the battery once your processor
demand drops.
These two facts conspired to drive me nuts. I'd get 'battery' problems while I was plugged in. . . .
Mine got fixed with a software update that 'reminded' my batter how to behave, and I then charged the battery fully so it could 'bridge' over the power deficit from the power pack.
MS suggested that SP3 owners could get a SP4 power pack, which allegedly has a higher power output to help . . . I did not do this.
My SP3 is working fine now, but I almost flung the thing more than once. . . .