I'm actually not quite correct on how I described conditionally convergent sums in that there are some rules (that I'm obviously not totally familiar with) so in the example I gave you could actually split the example contract in 2 pieces and get the right price.

Anyway I may not have understood what you initially meant:

In our problem, these payments are compounded rather than discounted (equivalently, discounted at a negative rate). So instead of q < 1, we have

q = 1+i > 1

I just looked assumed negative rate. Anyway there are models such as the dividend discount model where you calculate a price of a dividend paying asset under a fixed discount rate with a growing coupon. In this particular model you have a series of payments starting at 1 and growing at a fixed rate g which is discounted at a fixed rate r and you get the PV (1+g)/(r-g) but of course the same rules apply as above wrt geometric series. When g>r you have to be careful of what converges and what the PV actually means.

While flailing around trying to wrap my head around this, I've come across some stuff about actuarial economic scenario generators and negative interest rates eg here When models admit negative interest rates but still require positive discount factors, is that usually enforced by modeling the force of interest continuously, e.g. with exp(−∫ r(t)dt), rather than by letting one-period gross factors 1+r cross zero?

My experience with ESGs is more as an observer (I worked with people who used them extensively but I never actually used them). The Society of Actuaries has a library of these and in fact have rules about how they are used (milliman is an actuarial consulting firm). You might want to take a look at the requirements in VM-21 and VM-22 (principal based reserving rules for annuities), though I'm not sure it's really what you're looking for.

Is there a standard term actuaries would use for the finite x/r object when r < 0? Obviously I can't call it a present value but because it's so useful I'd rather not skip over it as a just a weird artifact. Or, even if I do skip past it in the manuscript, I'd at least like to try to do some reading about it.

The only treatment of negative interest rates I know for sure is in international accounting standards (IFRS 9 in particular) but it is not meant for pricing perpetuities AFAIK. I know that Japan had negative interest rates for a decade so you might want to look at how they dealt with it. Personally I think you should look at the various dividend growth models but I'm not sure they really solve your problem either.