As an update, the Lawson Ticket website for Ghibli Museum tickets does accept Monzo. However, they do a card check by doing a 1¥ transaction and it’s put through as a POS transaction so I got “Welcome to Japan” and “Enable magstripe” messages in my feed 
Protip though: the website was under super heavy load and was 503ing constantly. Took me an hour after the tickets went live at 2am to actually complete a full purchase. Persevere and you’ll get the tickets. When I woke up proper at 9am, most time slots were either sold out or had only a few left.