I'm hoping this new adaptor would do the job.
It's the answer you don't want - "It depends"
You need to find out what's inside the charger box itself, as the system you have will have used one of two designs:
Method A:
Use a power adaptor to generate low voltage DC, which then feeds a "smart" charger.
Method B:
Put the charge controller in the adaptor, and then the "charger" provides a physical interface.
Another thing to bear in mind, is exactly what is your charger providing? Is it providing regulated 24v or will the output voltage rise under no load.
Bear in mind that your proposed power supply is capable of supplying over 3 times the current. An unregulated power supply running at 1/3 load could have a higher that stated output voltage.