Dell Precision 5540 won't power on from new battery

Replaced my 56WHr original battery (down to half capacity) with iFixit's 97WHr replacement. Fits beautifully after removing hard drive bracket.

Laptop runs fine on AC. Sometimes when I unplug, it immediately powers off hard, as if no battery. Other times it runs a few minutes before powering off hard. And other times it runs for hours. All this from 85% or better starting charge.

Laptop will NOT power up from battery. First push of the power button, the power button, keyboard, and front power indicator all light up for about 2 seconds before turning off. No further activity. Subsequent pushes of power button yield nothing.

When I asked iFixit if it could be a bad battery, they sent a replacement. That one's behaving exactly the same.

I reinstalled the original battery, and it works as it should - short run time, but always holds up when full, and runs to empty.

Anybody have a Dell 6GTPY from here that actually works? Or experienced similar and fixed it?