My personal thoughts are battery or connector(s). Sure you've tried a couple batteries but there are duds out there...and if it's a "company" doing it who knows what they're installing...or doing...there's a reason I prefer to fix myself, because I know that work is being done right!...or at least know what work is being done :)
Anyways I digress...IF you're battery's life drops significantly over a short period of time, there's two options, the battery is a dud, or you have a SERIOUS short somewhere. Shorts can usually be identified by some sort of odd buzzing or excessive heat somewhere in the phone.
If you can 100% absolutely rule out the battery (I'm not convinced that you can yet because of that power drop you described), I would check every connector. Make sure they are clean and fully secured. If one is loose or dirty, just bumping or tilting the phone the right way can cause you to lose your power connection. Fidel the phone around and it reconnects and works fine again...until you bump or tilt it the right way again such that you lose power. Heck, just thermal expansion may be enough to break some connections.
Hope this can solve your problem, good luck!