If they were any good at troubleshooting they would have installed the new battery in the store, plugged a stock Samsung charger in and either using the Ampere app or a hardware metering adapter determined if the phone was now charging correctly or not.
I can tell you that I had problems with a Droid Turbo (the first generation Turbo), where if the plug was inserted but laid flat on the table it would charge but not at Turbo rate (fast charge). If I propped the cord to the charger up just enough to tilt the plug in the socket using my wallet the phone would then charge in Turbo mode. This signaled either a bad cable or bad socket. I tried all my cables and found several that worked correctly and didn't require then to be propped up but a number that had the same problem. The latter cables went into the trashcan and the problem went away.
I don't believe that the battery could have damaged the phone's charging system, and the fact that it charges fast sometimes points to an intermittent problem. Intermittent problems are more often than not mechanical in nature (i.e. failed cable), but an app could also be at the root of the problem.
A slightly bulging battery, as long as the case closes properly is not a serious problem but like a mole that you watch to see if it changes in shape or size signaling possible cancer, you should likewise take note how much it's bulging and see if it gets worse. Eventually it should be replaced but it never hurts to err on the side of caution except in respect to the expense of the new battery.
I wish I could tell you more but I still think the phone is not the issue. It is possible that the problem resides not at the point where the cable meets the phone but instead at the point where the cable meets the charger and that may be why moving the charger and phone to another plug (or house), thereby changing the cord orientation could result in it working at one place and not at another.
In order to properly negotiate Fast Charging there has to be "handshaking" (communication between the phone and charger), along the cable on wires and through connections that normally carry only data rather than the two wires which carry the current. An intermittent contact or break in a wire in the cable on one of those data lines, or likewise a wire or trace or contact that is broken in either the phone or charger could result in an interruption of the handshaking and prevent the two devices from communicating down the cable resulting in being unable to negotiate Fast Charging. As long as the two wires that carry power are still intact it will charge but only at the standard charging rate.
I would try different cables, different orientation of the charger and phone in relation to the cables, slight upward or downward pressure on the plugs, and see if you can get it to kick into Fast Charging.
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