Lattitude 7440 with erratic power behavior

I have a Dell 7440 with some erratic power issues:

1. When connected to the power supply, everything works well.

2. When disconnected it takes 10+ attempts to restart with the following behavior:

* during the boot process, the Dell logo shows up, just before the Windows login it goes dark again

* retry start, same behavior for a few times. After a few failed attempts the diagnostics kick in, but do not find something. ESC or wait to complete the restart sent the PC into Windows repair. Restart the PC sometimes works, sometimes does not. 

* after about 10 attempts it reboots

3. Sometimes the system has a power failure immediately when disconnecting from the power supply. Most time it holds for a while

I could not exactly reproduce yet, when and how it happens. It is erratic.

I did completely reinstall the PC a while ago, but it did not make a difference.

One possible root cause: I changed the battery and replaced a bulging original battery with an aftermarket battery. I believe that was when the issues started. I then replaced the battery again and it was good for a while. However, the problem started to happen again a few months later. 

Does anybody know, if this is a typical issue with Dell aftermarket batteries? I am wondering, whether the problem will go away with an original battery or whether there might be underlying HW issues, making a new battery a waste of money?

Thanks for sharing your thoughts