Oisin
(Apologies, I can't get the accent on my keyboard) -
I went through the London Gun Company to do the same last year and as a consequence I have contact details for an exporter in the US.
My cousin tried various couriers and the US Postal service to no avail. It ended up on a freight flight (not sure which carrier, but I will check), but it is not cheap.
In my case the rifle was owned by a late uncle who took it when he emigrated in the 70's, so the cost was irrelevant to me because of sentimental value.
It was worth @£200 in value and I think I ended up paying something in the order of £500-600 to get it here, if I remember.
Its the first (and only time) I have done this, so I was possibly (probably) 'striped up'. You will need to make up your own mind about value for money.
I also had to get an import code.
If it is of any help, send me a PM and I will root out the paperwork if you like.
Lloyd