Well, here is the original I got from the estate agent, so for those who want to challenge their translating chops, どうぞ！ I think many might conclude that he was not exactly clear himself, referring both to them issuing the 「外国人登録証明書」and the 在留カード, and he did indeed tell us it would take about a month to get the document we needed. As you don't actually speak directly to the scrivener until you're at completion, and the agent was reporting back to us what the scrivener told him was needed, it was definitely a case of Japanese whispers, rather than lost in translation.
Moral of the story, I think, is don't rely on your estate agent's real estate knowledge. Check for yourself. They're happy to take foreigner business these days, but that doesn't mean they're all up to speed with your special situation and can tend to the robotic.
Anyway, for those with a taste for Byzantine bureaucracy, enjoy!
"謄本の住民票に「property address」の住所が記載されていませんでした。
「外国人登録証明書」の追加取得が急務です。
東京都新宿区四谷1-6-1 四谷タワー13F, 入国管理局総務課情報システム管理室入国情報公開係
窓口の担当者に「jane doeさんの2000年1月1日から2012年7月8日までの外国人登録証明書のコピーを取りたいのですが」とお伝えください。支払いのため、至急必要です」とお伝えください。
在留カードの発行には通常1ヶ月かかりますので、最短で11月20日（月）に申請することは可能でしょうか？
おそらく11月24日の決済には間に合わないので、延長契約を結び、決済を延長する予定です。