That wouldn't solve the country's biggest problem: the huge gap that exists between the costs involved in raising children and the actual usable income people have.
I don't think there are any official figures on the subject, but it's fairly easy to find the results of independently conducted surveys where people are asked things like: "how many children would you ideally have, and how many are you actually planning on having?"
Here's an example:
http://www.hagukumi.ne.jp/angel110/special/21.shtml
In the first graph, 93% of the women replied that ideally, they'd like either 2 (45%) or 3 (47%) children, but 82% said they were only planning on having 1 (20%) or 2 (62%).
The reasons given are:
- Economical reasons (39%)
- Career (13%)
- Late marriage (11%)
When asked what kind of solutions they think would be efficient in fighting the decline in birthrates, their answers were:
- Economical support (40%)
- A more parents friendly work environment (12%)
- Easier to enter childcare (11%)
And these are 2006 numbers. The situation has actually worsened since then (with the recent suppression of big tax deductions for couples with children, replaced with nothing but tears and cries).
Immigrants would have exactly the same problems.
The only immigrants who would be willing to have more than 2 children in current Japan would be those who aren't planning on putting a lot of money in their children's education.
It is probably safe to bet those would be mostly unqualified workers.
But Japan doesn't have enough of a need for unqualified workers to justify massive immigration.
The country already opens its doors to foreigners in sectors where they are needed - nurses (to take care of the elderly), for example.