I do not know what will happen if the battery is removed - I will try to find out for you.l
You should be able to get a lithium battery of a suitable type from most electronics places - in the uk it would be RS components , farnell (Newark in the states I beleieve), etc - package type doesn't matter as you have it off the board already - just make sure it is the right chemical type (lithium) and 3.6v.
Once you have one you could do the following to be safe - on the wires leading from the battery scrape a little insulation away on each one and solder the new on ein across the leads .... DO NOT DISCONNECT the old battery until you have done this ! Heatshrink or insulate where you have soldered to prevent shorts.
Once it is all done snip the wires at the case of the battery - this will ensure you have a constant voltage throughout. Insulate the newly snipped end (heatshrink, etc ) and you are done.
In about 2 to 3 years uninsulate the cut ends solder a new battery in and desolder the other one. Insulate the small batch of bare wire.
Repeat every two to three years !
Iain