That they believe that slavery can be replaced by a more 'correct' form of labour, machines, and starts to recruit/attract all those mechanics and engineers? And simply push for more research and to build reliable steam engines that can replace alot of the heavy lifting work.
That's not terribly Stoic. Stoic philosophers by and large believed that humanity was a unifying bond between all men, including slaves, so slavery was something vaguely distasteful to them. But the idea that you need to remodel the state to accommodate this idea better is a bit out of whack with a school that teaches fate must be accepted and dealt with in the most dignified manner possible. The laws that eased manumission and improved the lot of slaves are much more in keeping.
More to the point, though, I think slavery is a red herring. Marxist and Western historians alike have been able to grasp it as a simplistic explanation of why the Romans were so shockingly unvictorian. A lot of research has been done since, though, and we tend to find that we really don't understand slavery very well - with the exception of late Republican Italian slavery. That, unfortunately, evolved under exceptional circumstances and can not be extrapolated from.
If Rome had wanted an industrial revolution, it would have been doable. The main problems here are neither slavery nor technology. Roman slavery could have accommodated industrialisation fairly well, with the verna system and manumission for highly trained staff as an incentive to perform. Technology could have provided machionery easily enough - it was fashionable for a long time to denigrate Roman technology and trade, but archeology is so full of publications showing the exception that we are coming around to understanding the exception is the rule. The Romans were good at technology (There were big, probably insurmountable, obstacles to building viable steam engines, but they are the same that would have prevented building a computer in 1820. It would take time is all).
The problems you'd need to solve are, in no particular order, the absence of complex financial thinking and applied higher maths, the relative dearth of raw materials, limited written communications, dearth of technically skilled labour at the base and heavy transport infrastructure. All of this is doable.