How can a free market be operated in a way that significant social and environmental costs are intrinsically covered?
As far as we can tell, no, there is no way to do it using only a free market.

Coase postulated that within a free market, agents would negotiate private contracts to internalise the externalities.

In reality, this didn't happen. It turned out that humans aren't perfectly informed hyper-rational beings. And as Coase himself had inferred, the transaction costs are typically too high.

There are well-documented ways in which markets have been adjusted for externalities: sometimes taxes or subsidies are sufficient; other times, prescription or proscription has been necessary.

There are even well-documented ways in which the commons have been protected, meaning that the destruction of them is no inevitable tragedy: Elinor Ostrom won the Nobel Economics Prize for her work showing how empowered stewardship by a group of custodians had repeatedly protected the custodians' commons.

So the relative efficiency of market mechanism versus planned mechanism is nowhere near clear enough to claim that "It would be far more efficient and sustainable if the tendency to externalize costs were managed by market mechanisms ... ". Each of the alternatives has inherent inefficiences: planned mechanisms have dead-weight costs. market-mechanisms have transaction costs and failures due to imperfect information and bounded rationality. There's a lot of evidence, particularly when it comes to the wicked problems you describe, that the market inefficiencies are far worse than the planning inefficiencies.