I am a University Postdoc (got my PhD a couple of years ago). Many of the fellowships/grant applications require that the new "sponsor" is different from my line manager. The line manager has the tendency to want to keep me and suggested that in case I apply with someone else and the proposal doesn't go through "there will be questions in future applications if we are seen to keep working together after your current application". This could be a genuine concern, but also sounds a bit like "if you apply with someone else we're done". You'd expect that your line manager would just want you to grow irrespective of the path you chose, but we all know life is not ideal. How to deal with such situations? Should one apply without informing the line manager? It sounds strange as you would probably need their recommendation letter. Should one simply start applying to lots of venues hoping at least one goes through, knowing they won't be able to go back to their current line manager? Can you all share some experiences from your labs on this topic? Thank you!
I don't know your postdoc supervisor/line manager but I doubt that their advice has the subtex of "if you apply with someone else we're done." Hopefully you have a good relationship with them and can discuss it with them. A good supervisor mentors their people to become good independent researchers themselves. I see it as training future colleagues and friends, not competitors.
I think their recommendation, to work with a new sponsor, is good advice. I would also advise postdocs which I supervise to do the same. This is because it is difficult to establish credible independence if you continue in the same research group. Working with someone else will also broaden your network, which is important for an academic career.
Some fellowships will explicitly forbid working with the same supervisor. For example the Imperial College Research Fellowship sponsorship requirements state:
Applicants from Imperial are required to move from their existing sponsor/supervisor to a new sponsor for their ICRF, to increase both their independence and their breadth of experience
Other fellowships may not explicitly state this as a requirement but could very likely view moving to a different supervisor as a positive.