Phone powers off until battery is disconnected and reconnected

I recently replaced the battery in my iPhone 5s with a part form iFixit.

Since then, on two occasions, the phone will power off without warning, even with a full charge and won't boot up again.

I discovered I could fix this by opening up the phone, disconnecting the battery, then reconnecting it. When it boots up again, everything is normal.

One thing I did notice is that if the phone powered off with a full charge, after I dis/reconnect the battery it comes back on at around 60%, as if it's been using the battery.

It's not just a display issue - the phone doesn't make noise or vibrate when I get texts. When I open the phone, the battery connector and the screw-on cover plate are still intact.

Anyone have a clue what could be going on?