If I settle my dispute with the seller privately, is that settlement enforceable like a court judgment?

Yes, generally — a properly documented private settlement, such as a signed settlement agreement or minutes of settlement, is a binding contract in its own right and is generally enforceable like any other agreement between the parties. If the seller then doesn't honour it, you can typically sue for breach of the settlement agreement itself, which is often simpler to prove than the original underlying dispute, since it usually just requires showing the settlement's terms and the failure to comply with them.

Where litigation was already underway before the settlement, some settlements are also built to be filed with, or incorporated into, a court order, which adds a more direct enforcement route beyond an ordinary breach-of-contract claim. What actually makes a settlement enforceable is having it properly documented in writing with clear, specific terms — an informal verbal understanding between you and the seller is considerably weaker and much harder to enforce if a dispute later arises about what was actually agreed.