We just completed a similar trip. Your chances of getting into the Private room are good, but not for sure. We were staying in the transit hotel in the airport airside, and this fact seemed to sway the hostess at the Silverkris Lounge, who escorted us to the PR the evening before our 9am departure for NRT and LAX.
Your chances of getting into the Star Alliance lounge (which Singapore uses)
at LAX are quite poor. It will be manned by SA personnel, and the rules for that lounge explicitly state it is for departing F and J passengers only, and for other elite members, lounge members etc. Never hurts to ask, though.