If your FAC is conditioned to allow you to use the .308 for deer but not fox and you are experiencing resistance from the police to allow that calibre for other quarry, then you might want to draw their attention to this (from BASC):
The Association of Chief Police Officers (ACPO) Firearms and Explosives Licensing Working Group (FELWG) has issued advice that supports Chapter 13.14 & 13.22 of the Home Office guidance. It advised police forces to allow larger calibre rifles to be used to take lesser species i.e. where the primary reason for possession e.g. deer stalking was established, all lesser species such as foxes and pests could be shot. This practice has been commonplace in Scotland for some time, and has not endangered public safety in any way.
In June 2009 ACPO FELWG again advised forces of a new condition that can be used in place of the standard condition in Appendix 3 of the Home Office guidance.
The (rifle/sound moderator/firearms/ammunition) shall be used for shooting (Named Principal Quarry Species) and any other lawful quarry, on land deemed suitable by the chief officer of police for the area where the land is situated, and for zeroing on ranges, over which the holder has lawful authority to shoot. (Delete italics where appropriate).
This identifies the primary reason for use e.g. deer/fox control, then allows “other lawful quarry”.
If on the other hand you are saying that you already have a certain piece of land cleared for .22RF, and they won't allow .308 as you are restricted to specified land only, then it's a different kettle of fish. The Home Office Guidance (13.11) states that where possible the applicant should be present when an inspection takes place in order to demonstrate that they understand the characteristics of the land and the best places to shoot safely on it. You could request to be there and put your case directly to the FEO.
If all else fails and you think they are wrong in their appraisal, (is the FEO in anyway qualified to undertake inspections?), you could request the reason for refusal in writing - it might focus their thinking - and take it from there.
Have a look in the Legal Issues section of the forum for other information on the subject.