...There's no connection to speak of between the impacts of carbon emissions (SCC) and the estimates of the remaining carbon budget...Which made me think. Hey, that doesn't happen often. So, yes, there is no connection between SCC and carbon budget. Because the two ideas are different: SCC is "how much damage does emitting this CO2 do?" whereas a carbon budget is "how much CO2 can I emit before I do any damage?". That's not quite right, I have simplified for effect, to enhance the separation of the concepts. For a carbon budget, you've decided on an acceptable level of temperature change, perhaps 2 oC, and guessed your climate sensitivity, and so decided on an allowable amount of CO2 to emit. If this were, truely, a budget, which you weren't allowed to outspend, then your cost of emission would be zero, up to the limit, and infinite beyond that.
And so of course we're back at the carbon-taxes-versus-carbon-emission-permits debate again. Because the budget stuff is analogous to the permits.
The budget stuff has several big problems, which one study touches on. I'll ignore the quasi-arbitrariness of the size of the budget. The problems are that the budget is global, in space and in time. So no particular place, and no particular time, have any incentive to minimise their emissions. Unlike - ta da - a carbon tax, which conveniently provides incentives, and copes well with uncertainties. The disadvantage of the carbon tax, of course, is that it doesn't invite endless chatter in place of actual activity the way carbon budgets do.
I think you can be stronger than "no connection" between SCC and budgets: that if you're keen on SCC, you have to use that; you can't use budgets as your tool. They become only a useful (perhaps) illustration. Or conversely, if you really want to use budgets, then the implied-equivalent-SCC is just an illustration.