I would not say it is messy. Since the pulley's velocity does not change, there is a simple equation relating that impulse to ##\int T.dt##.Steve4Physics said:A rigorous analysis (i.e. not using any ‘tricks’) would need to include the resulting external impulse, which sounds messy.
But I would like to expand on my comment (post #10) about the flaw in the question. Inextensible does not imply inelastic; in principle it could be perfectly elastic.
My approach to such idealisations is to take them as the limits of a more realistic set up. Here we could suppose some elasticity and a small extensibility, analyse that in conjunction with a completely inelastic collision between the putty and the mass it hits, then take the limit as the spring constant tends to infinity and the elasticity tends to perfect.