Hello @jirib and welcome to the forum!
I haven’t set up WinPE, and unfortunately this question doesn’t seem to have come up before in this forum or GitHub issues, so I guess I’ll see what comments I can make. Perhaps someone more expert will come to help.
For a double bonus, I’m not familiar with burp either.
On Windows, Duplicati requires at least .NET Framework 4.5. Microsoft documentation refers to some subset version that’s available (I have no idea if it’s enough for Duplicati use), but other folks add in what they need. Adding something recent would be best. I’m not sure when 4.5 support will be dropped, but it’s growing old…
Although Duplicati has a web UI, possibly all you need is command line, and it may avoid some WinPE setup. Routine backups will be fastest (and work best) if Duplicati backs up from the source computer to destination. If that’s your intent, then recovery basically means having the source know its configuration, e.g. by export of configuration from the web UI or some other means. Note that the configurations contain secrets which might be tailored per-computer or shared if that’s OK. –parameters-file might be helpful some place in this scheme.
Having a remote Duplicati server scan the source host for changes would add drawbacks, but helps secrecy and also avoids having to recreate the sometimes troublesome local database if the source host suffers loss. Server would still need protection though. A third-party Duplicati client could be used to talk to remote server.
I’m not sure if either burp or Duplicati can restore the whole Windows machine as nicely as an image backup, however Duplicati has a lot of advanced options that might let you tailor your design to its best possible form.