That could be because we 'have' (auxiliary verb) is different to we 'have' (verb meaning 'to possess'. We only contract the form with the auxiliary verb in this case 'we've done it already' = 'we have done it already' and not with the verb for possession, so 'we have a cat' and not 'we've a cat'.
Yeah I'm not 100% why (that it's not an auxiliary is still contracted). But I found this on wiktionary:
'The have in have got is almost always contracted (e.g. I've got, he's got, John's got). In the sense of to be obliged, the have is sometimes not contracted when got or the subject is stressed in the sentence ("I have got to go there." and "I've got to go there." are both common but "I've got to go there." is almost exclusively preferred in spoken English over "I have got to go there."). The uncontracted form is considered to be formal in Polish teaching of English. '
'The have in have got is almost always contracted (e.g. I've got, he's got, John's got). In the sense of to be obliged, the have is sometimes not contracted when got or the subject is stressed in the sentence ("I have got to go there." and "I've got to go there." are both common but "I've got to go there." is almost exclusively preferred in spoken English over "I have got to go there."). The uncontracted form is considered to be formal in Polish teaching of English. '