According to the Embassy of Japan in PRC, the length of stay of tourist visas is not predefined unless you are travelling with a tour group. The Embassy asks you to contact a pre-approved travel agent, which handles visa applications for you. To minimize risk (the travel agencies are supposed to pre-screen applications), some travel agents might place arbitrary limitations on your stay. However, if you go to a large and established travel agency, such as the China International Travel Service (中国国际旅行社), they likely would help you get a visa for stays of 30 days. From personal experience, I am very certain that one person I know of (my mother) has been approved for a 90-days visa, and many other people I know have been approved for 30-days. So in terms of possibility: yes, it is definitely possible.
So, you have two options here:
when you arrive in China, apply for a visa at a CITS (or other major travel agency's) office. Prepare her household register (Hukou), bank statement, salary sheets, etc.
if you (and your GF) have a decent income, she can try to apply now for a multiple-entry visa. Depending on where you live, you need to contact the visa section of either the Embassy of Japan in the UK (in London), or the Consulate-General in Edinburgh, to ask for the qualifications & documentations required for the visa. A quick Google search will tell you the email address.
I'd go for the second option because: (1) she gets a multiple-entry visa, and (2) she might have problems finding proofs of her resources in RMB (the agency & consulate might not accept bank documents/salary sheets in GBP). But then it all depends on the circumstances.