The JAL first lounge gets busy at the best of times, if they’re renovating and have chunks of the lounge closed then I can see why they’re turning people away. It would be quite unusual in normal times but when lounges are running at significantly reduced capacity due to upgrades it can happen.

If JAL First is off the table then AA is probably the best lounge, the QF lounge is an OK business lounge but 10 years old and due for a refit. I really think going into Narita is the best option, from my read the OP likely needs to come landslide to change terminals anyway because I don’t think there will be interlining with those airlines so I’d be leaving luggage in terminal 2, going to Narita for a couple of hours, coming back, going to one of the lounges for a shower and drinks then board.