Hi Benny - I went through this about 18 months ago. My job sounds rather different to yours but I found the Direct.gov website absolutely brilliant.
From memory (and this may have changed), they can only turn down flexible working requests on 8 different grounds and they have to specify which ones in any refusal. The only one they really could have got me on were the cost implications (I have moved up to the NW and my office is in London so I have to catch the train a couple of times a month) but it sounds like they are trying to get you on the customer-facing bit of your job.
Anyway, in a way I was lucky because my vindictive ex-boss (who herself was wfh from Belfast!
) wrote a refusal letter that wasn't based on any formal reasons but a lot of hearsay and what other people did, how much they cost etc etc. I was able to write an 11-page rebuttal of all her points in writing which formed the basis of my appeal.
I could go on but obviously I don't know your situation. The appeal meeting was rather charged but it was carried out (as per the rules, again see the Direct.gov website) by an independent Director with HR sitting in to observe and I took a manager from another team i.e. someone I didn't work with but who I vaguely knew was fair and good to her staff. I briefed her fully beforehand but she was basically there to take notes for me as I didn't want to have to make sure I was getting everything down or anything was being twisted. We had a quick debrief afterwards as well so that what I thought had been said had (or not).
Here are some pages you may need, if you haven't seen them already. My tip is to be as completely prepared as possible and try to anticipate their sticking points. They should have given you something in writing to work with. They also have to give you 14 days for the appeal meeting and you have to agree to the date so don't let them pull you in at a minute's notice, make sure you're ready and your rep/friend.
Dealing with the outcome of your application www.direct.gov.uk/en/Employment/Employees/Flexibleworking/DG_171775
The important info for Employers on refusing a request (you may be able to get them on a technical basis if they haven't followed this)
www.businesslink.gov.uk/bdotg/action/detail?itemId=1097377943&r.l1=1073858787&r.l2=1073858926&r.l3=1073931239&r.s=sc&type=RESOURCES
Info for Employers on an employee's appeal
www.businesslink.gov.uk/bdotg/action/detail?itemId=1081563827&r.l1=1073858787&r.l2=1073858926&r.l3=1073931239&r.s=sc&type=RESOURCES
You may also find help on MN in the legal section?
Needless to say I won my appeal and I now work from home... which is why I'm still working on something urgent at this time of night!
The very best of luck. :)