Hi there,
I did Further Maths GCSE and can confirm there isn't much additional stuff in the GCSE, the majority of it is standard maths GCSE, and we were taught it after school once a week for three months (so at most 12 hours of additional content), so it should be very easy to self teach if you wish!
Although your school doesn't "offer" it, they should still be able to enter you for it, it's as simple as entering you for any other qualification that they do teach you, apart from they just need to know you want to do it (inform the exams officer well in advance (probably January Y11) so that they have record of it) but there should not be any issue there, they should be more than happy to enter you, as it is their job to do so!
Hope this helped!