HTTrack does not emulate user actions such as clicking buttons. But, you may
can experiment with the information given here:
<http://httrack.kauler.com/help/Form-based_authentication>

It describes how to capture a URL, for the scenario of a login which is
form-based submitted.
In your case it seems also form-based, just that it is not a login, but a
simple button click. You could try to simply use the URL afterwards as
displayed in the browser, but I am afraid it is using cookies and then this
simple approach likely won't work. However, perhaps the example description
for login perhaps works out for you still.

Do you use Windows + WinHTTrack or command line version of HTTrack? The
description is for WinHTTrack GUI version, however, technically it is also
possible with command line as per my understanding (though I haven't done it
yet myself).

As I don't know if the website you mention is legitimate and since I don't
understand Japanese, I can give you only general hints as mentioned, but won't
try it myself.

When you search this forum for keywords such as "button click", "form submit",
and the like, you find perhaps further helpful information.