It's best to mail these questions to "registrations AT
planetside.co.uk". No Oshyan, I'm not applying for any of your jobs
lol
They will then probably send you (your key if necessary and) the link to download the latest version.
When you use the update-function in the beta and it detects a new version you're being redirected to the downloadlink.
So you can always install the beta on your crappy computer, check for update, download it and then copy that part to your TG2-computer.
At least, that's what I would do.
Anyhow, e-mailing Planetside is the best thing to do I guess.
Martin