I wouldn't attempt it on separate tickets at the moment. Things can change between now and then but if you present for your initial flight JAL may recognise you as a Japan only bound passenger and thus deny boarding if you're not eligible to enter Japan.

As for the taxes, it is probable extra taxes are added because you are no longer a transit passenger but have two seperate tickets meaning surcharges/taxes/fees are added for your arrival and departure.

BA.com is not perfect for booking reward flights, and if it's not a married segment issue, an agent may be able to book it as a through ticket. I had a similar issue with a QR ticket a few days ago.