General information, not legal advice. Three layers worth keeping separate. First, ownership between you and your client is set by YOUR contract with them, not by the tool. A normal work-made-for-hire or assignment clause passes your rights to the client, so the ownership question is mostly about having a clean contract. Second, your rights against the vendor come from the tool's terms, including any indemnity, which as noted above tends to be conditional on settings and plan tier. Verify the live terms, don't rely on forum memory, mine included.
Third, the copyleft concern is real but manageable: keep duplication filters on, run a license scanner over the codebase like you would for any dependency audit, and disclose your AI-assisted workflow in the engagement so it's a known, accepted process rather than a surprise. Most disputes I see in this area come from surprise, not from the code itself.