I don't like using LiPo batteries as the top end voltage is rough on the voltage regulator built into the transmitter. Using them voids your warranty (not that yours is under warranty) but it shows that Spektrum does not recommend them either.
What I like is LiFe batteries. They are lower voltage so a 3s pack will output ~9.9v perfect for a transmitter from 8 NiCad cells.
Does your new charger support LiFe charging? If so that is a great option IMHO.
Here is a pack that will fit.
You will need to replace one of the Futaba connectors with the old connector from your NiCad pack.
If you are OK with ordering from HobbyKing in China the sell packs with the Spektrum type connector already installed.