We do find, however, that the recycle bin is always overflowing, and quite often end up putting the recycleable rubbish in the black bin just to stop it overflowing.

We had this problem too.
If you contact the BCKLWN and complain to them about it they can do one of two things. Either issue you a second green bin, or remove the 240 and replace with a 360 litre green bin.
In true council style, someone has to first come out and rummage through your green bin to 'make sure' you are not placing in there anything that is not on their list of accepted materials*
*This list is quite restricted, so if the bin-rummager finds stuff in your green bin that is not on their list they will probably try to say that you do not need a bigger bin.
Basically, what I am saying is, if you do arrange a visit, 'be prepared'.