A quick Google search shows that the 450d uses an LP-E5 battery, also used in the 500d. Since the 500d was current until 2009/10, I'm sure Canon still has spare batteries available.
3rd party vs original? I've used 3rd party for years without any issues, but some others report shorter lifespans and other probs. More recent Canon models used 'intelligent' batteries, and don't show the remaining capacity of 3rd party batteries, but that shouldn't be a problem for a 450d