I'm not sure why the default position here should be to accept that somehow the little ice age had a causal effect on industrialisation.

For one thing, there is no obvious underlying theory; how does a variable-but-persistent cooling in the centuries preceding the industrial revolution lead to technological change? One might theorise from things like the demand for heating fuels and the exhaustion of wood supplies, or some kind of relative productivity effect with agriculture. But this would be a very vague set of speculations, not a solid theory of the industrial revolution.

For another, the chronology is not a very close fit. The little ice age (presuming its existence) lasted for centuries, beginning around 1400. The classic dates of the industrial revolution are over three hundred years later, right at the end of the LIA. Why would a downturn in the weather take over three centuries to trigger an industrial revolution? If it is only the rough chronological concurrence of the two events, why focus on those two events, and not the innumerable other factors?

There is a general literature on the "general crisis of the seventeenth century," a supposed synchronisation of upheaval across Eurasia, and there is a book by Geoffrey Parker, Global Crisis: War, Climate Change and Catastrophe in the Seventeenth Century, that posits that the LIA played a causal role in this process. (I have not read the book, though I did see Jan de Vries discuss it in substantial detail, as in his review here.) Some of these events, perhaps the ones centred around the North Atlantic (the English Civil War(s), the Dutch Revolt, The Thirty Years' War, the Glorious Revolution, etc...) might have had some kind of effect on the industrial revolution. Certainly there are theories that suggest some of these events had an effect on the IR, like North and Weingast's famous paper on the fiscal effects of the Glorious Revolution. But this is a theory requiring at least two dominoes, where climate change leads to political upheaval, and that political upheaval leads to the industrial revolution.

There has been some debate about to what extent the LIA is real, or relevant. Cormac O Grada and Morgan Kelly rather boldly suggested that the whole notion of the LIA was nothing more than a statistical artefact of smoothing (the Slutsky effect, where smoothed random numbers create illusory "cycles" out of white noise). They had a back-and-forth with Sam White and with Büntgen and Hellmann in the Journal of Interdisciplinary History. A more recent econometric analysis by Damette et al. using Markov switching methods seems to suggest that there is a discernible cooling pattern above and beyond the Slutsky effect illusions suggested by O Grada and Kelly.

I'm not sure I quite grasp whose views are correct; I understand the basics of how the temperature proxy data sets are constructed, and the idea of a Slutsky pattern. The Maunder Minimum is clearly a real solar phenomenon, though it does not start the (assumed) LIA. And the North Atlantic Oscillation pattern clearly can cause climactic variation. But the econometrics of the more detailed analyses rapidly go above my pay grade. For whatever it's worth, I currently believe this is a real phenomenon, but that the effect is subtle and extremely variable by year and region, and therefore almost impossible to use for causal attribution. Perhaps this might work for things like "cold weather reduced crop yields," but "cold weather caused the industrial revolution" seems well beyond our ability to discern.