Agreed, keep the connector! Although if you are gonna cut up the old battery, you could just cut through the foil of the battery and glue the new one on the foil of the old one.
On a side note: on Aliexpress I bought two
plastic open-up-your-mobile-set for 60 cents or something. They break just that bit earlier than your gadget. I got the hang of it early, so I didn't need more than 2 sets