> add a back door?

What's your threat model here? Some kind of mass hacking attempt? It would be easier to attack the service providers, rather than steal legitimate logins.

A targeted attack on a specific person? It would be easier to, as the famous XKCD suggests, drug and/or hit them with a wrench until they voluntarily hand over whatever information you want.

It's difficult to conceive of a situation where hacking password managers is the path of least resistance.