You have to maintain a fertility rate over 2 to maintain a culture.
Nope. Australian tribes with a population of just 80 to 100 people were able to maintain their culture for tens of thousands of years, until the arrival of the Europeans. These tribes had unique languages and customs, and despite such a minuscule population, they were able to survive until the late 19th century. Normally, anthropologists consider tribes with a population of 400 or above as "viable".
At 1.3 a population would be cut in half in 45 years an in 100 years would be down by 75%. No kids means decline in population, plain and simple, people die every day.
If the Japanese don't want to have children, then what can the government do? And unlike the Western nations such as Sweden and Denmark, the Japanese government does not provide much maternity and paternity benefits to the young couples.