Eh? Why the fuck does having a particular phone matter? :S
I got that problem when trying to access Planet 3 originally - since my Nexus S isn't an official handset - but didn't care much since I assumed it's boring crap there anyway - but at least with that I can see why they'd ask for your phone, to only show stuff that works on it.
But having to have a particular model to remove the block is just... wtf?
Yeah, I'm wondering if Blackberries just use the other APN, even on PAYG, so maybe you could try changing that and seeing if it works for you?
I guess there's a chance they'd apply extra charges for doing it, but if you felt like risking that you could set your APN to "3internet" (instead of "three.co.uk") and see how it works.
(There's possibly more settings to change if you use MMS, but I think just setting the name and leaving everything else on default/same as other APN is good enough.)
Costs a fiver a month to use Blackberry services I imagine that means they have a BB Enterprise Server setup to handle internet/email/PIN messages etc, so wouldn't work with a non blackberry. I'll see what happens with the apn though...