Bells theorem rules out only local hidden variables. Quantum mechanics itself works around the issue not because of indeterminism, but because wave function collapses everywhere at once in a non local way.
But perhaps we do not even need to abandon locality, if we modify it a bit. There is no good reason to believe that space at short distances should be similar to Euclidean space, one interesting hypothesis is that space is more like a graph, and entangled particles in addition to the normal long path through the graph are also connected directly, which allows measurement on one of them to change the state of the other.
I'm not a theoretical physicist, but I've heard in informal conversations with some that one idea being explored more now is that perhaps space itself is just a statistical emergent property of entanglement.
Leonard Susskind has several interesting lectures about it which can be found by searching for ER=EPR, but i hope the final theory would explain more of the strange behaviors of quantum mechanics, something like the theory outlined in https://blog.stephenwolfram.com/2015/12/what-is-spacetime-re...
Yes, the idea is that the particles themselves are connected to each other like two ends of a wormhole, so the signal about applied measurement doesn't have to take the 3km long path outside, but can directly go from one entangled particle to the other.
Entanglement would allow to send information if it was possible to clone quantum states https://en.wikipedia.org/wiki/No-communication_theorem#Some_.... If we are looking for a hidden variable theory, we need an explanation for no-cloning theorem independently from entanglement issue, so this does not add any additional restriction.
But perhaps we do not even need to abandon locality, if we modify it a bit. There is no good reason to believe that space at short distances should be similar to Euclidean space, one interesting hypothesis is that space is more like a graph, and entangled particles in addition to the normal long path through the graph are also connected directly, which allows measurement on one of them to change the state of the other.