Tax geolocation is based off their invoice address and specifically the invoice address attached to the basket they're paying. Is that not working at all?

I'm guessing because they live outside the taxed area their invoice address would be different from their taxed address so you'd probably needing 2 different addresses for this situation then? I think you could do this by having 2 CB fields where on profile they can specify their tax geolocation or maybe even just a field to select their organization and branch then use that instead. Basically they can supply their country and state independent from their invoice address. Next you could create a negative promotion in CBSubs Promotions that's applied if that field has a value (use Conditions tab to check the field for a value and what the value is). You'd then apply the tax using the negative promotion (e.g. -3 percent discount would increase the price by 3%)

So does the invoice address get used in the tax calculation at all, or is that just for the printout receipt?

The invoice address is passed to the basket and that is then used for tax calculations. So yes it's used for tax calculations.

As there are 27 rules for calculating tax, I need the system to do it as opposed to a fixed rate.

I suggest they just set their invoice address to their tax address. If their tax is based off an organization then in theory their invoice address should be that of the organization. You could even change the language string so it's titled "Organization or Branch Address". Next configure your payment gateways to not pass the address to the gateway since it's not relevant for the payments. Your profile fields would then just be used exclusively for their personal address.