Learn Spanish (Latin America): En cualquier caso, hay que avisarle.

They are related, but serve slightly different purposes. While ahora bien (however) and no obstante (nevertheless) introduce a contrast that directly opposes or adds conditions to the previous statement, en cualquier caso (in any case) is used to dismiss or brush aside whatever was just said. It shifts the focus to the inescapable conclusion: regardless of what was mentioned before, warning him is necessary.

The translation says 'we have to let him know,' so why use hay que instead of tenemos que?

Hay que expresses an impersonal obligation, literally meaning 'it is necessary to' or 'one must.' While tenemos que avisarle specifically means that 'we' have to do it, Spanish speakers frequently use hay que to state a general necessity. In English, we naturally translate this general need as 'we have to' or 'someone has to' to make it sound natural.

Why is it avisarle with the indirect object pronoun le, rather than the direct object lo?

The verb avisar (to notify or warn) generally takes an indirect object for the person being informed. The direct object would be the actual information given. Think of it as giving 'a warning' (the invisible direct object) to 'him' (le, the indirect object).

Why is it cualquier and not before the noun ?