Here's a different approach, albeit less efficient, that only uses things that seem to be working for you. I'm not in a place where I can test code, so I'm just going to sketch this out in terms of ssh commands. You are trying to connect from A to B and then from B to C:

ssh -t -l usrB machineB
ssh -t -l usrC machineC uname -a
[download]

Instead, try connecting from A to B and creating a tunnel from B:2222 to C:22. Then connect from A to B:2222:

ssh -f -N -l usrB machineB -L 2222:machineC:22
ssh -t -p 2222 -l usrC machineB uname -a
[download]

A key difference here is that the main process has to start the first ssh process and then resume execution while the ssh process continues. I've never tried this from a script, so it is possible that it requires a fork. If this does work, you can reduce the added overhead by choosing weaker encryption for the tunnel's data and rely on your second connection for security.