To me, this feels like a "trailing though..." we sometimes put at the end of English clauses / sentences, to imply that there's more left unsaid, and what's left unsaid is typically a request or invitation.
聞きたいんですが => "I want to ask, though..." お借りました車、調子悪いんですが => "The car I rented has a problem, though..." ちょっとお願いがあるんですが => "I have a small favor to ask, though..." 銀行を探してるんですが、近くにありますか => "I'm looking for a bank, though... is there one nearby?" 少し、休みたいんですが => "I'd like a rest a little, though..."
