Breakdown of Powinniśmy byli ich uprzedzić.
być
to be
ich
them (virile)
powinien
should, ought to
uprzedzić
to warn (give advance notice)
Questions & Answers about Powinniśmy byli ich uprzedzić.
Why do we need byli here instead of just saying powinniśmy ich uprzedzić?
Powinniśmy by itself simply means "we should" and refers to the present or future. By adding the past tense helper byli, we create the past unfulfilled obligation "we should have". This structure makes it clear that the obligation is in the past and we didn't actually warn them.
How do I know whether to use był, byli, or były after the powinien form?
The past auxiliary has to match the gender and number of the subject. Here, the subject is "we" in the virile (masculine personal) form, which is why we use byli. If the speakers were a group of women, the sentence would be Powinnyśmy były ich uprzedzić.
What case is the pronoun ich taking here?
It is in the Accusative case, acting as the direct object of uprzedzić (to warn someone). For masculine personal pronouns like oni (they), the Accusative form ich happens to look exactly the same as the Genitive form.
Could we use the imperfective verb uprzedzać in this structure?
Only if the meaning were different. We use the perfective uprzedzić here because we are talking about a single, specific warning that failed to happen. If you meant "we should have been warning them" on a regular or continuous basis, you would use the imperfective uprzedzać.