I'm sorry that just comes across as unprofessional, weak and passive aggressive. If someone started doing that on my team I'd take it as part of the case against them not a reason to fight to keep them. Also presumably that's in ear shot of other team members, it's disruptive to team morale. If you are serious about looking elsewhere, make it clear you want to stay but xyz is making you consider other options. Do it in private with the right people. Or say nothing at all.

Very principled of you. Not sure management would be as ethical.