If I did not understand your question, explain it in more detail.
WHY for example do you need to connect a charger to "power it on" when the UPS itself should be charging up the battery.
If the device isn't working then the battery is either dead or the UPS isn't charging it (or other issues).
*To be clear, if you end up trying to draw a LOAD from the 12V charger (since it's in parallel with the battery) there's a good chance you will fry the charger or it may have protection to simply shut itself off.
I think you're trying to use the charger as a secondary battery but it's almost certain that will not work. Again, don't mess with the stuff if it's not 100% clear.
Replace the battery, or entire UPS if it's not working as it should.