Can the Qing keep their throne with a POD after the First Opium War? Or is their demise terminal?
I'll define it as retain 75% of their post-Opium War territory and maintain power on their throne. It can be an absolute monarchy like the've had, or at least an executive monarchy (a litmus test is have at least as much power as James II of England pre-overthrow). Since they don't really have a tradition of nobles gathering in a group and having power (like Parliament or the HRE Diet) then probably "absolute monarchy" and "lose the throne" are the only possible outcomes.
The problem is that the defeat against the British was pretty damn humiliating. It looks pretty darn terminal to me. Much like how the Sack of Rome in 410 was pretty insignificant materially (it wasn't their capital and the wealth looted was less than 1/30 of what they were losing to raids already) but legitimacy was lost.
It doesn't really have to end up as a Great Power to meet the challenge. For example, sucking up to the British after defeat in secret, fighting internal corruptions, making a small industrial area the size of France with textiles chemical factories or whatever, stopping the famines, and controlling internal policy while stabilizing their country fits the challenge, even though this leaves them unable to influence the world stage militarily.