Just to update the thread with my experience on the trip, I did take a taxi from HND to the Conrad, making my wife very happy. Taxis were plentiful when we left the airport (around 6am) and the cost was ~7700jpy. I was very happy with that decision when later that day I took a train to Shinbashi and then tried to walk to the Conrad. My first time through, I found the station challenging to navigate and the route to the Conrad somewhat confusing- I was very glad that I wasn't trying to find my way with my luggage. Once I walked it in the reverse from the Conrad to Shinbashi, I realized how simple and short the route actually is, but for a first-timer, the limited and changing signage made it tricky and the taxi was a stress-free experience.