Passing a long string from SoftwareSerial to Serial
The output of the logic analyser is pretty clear. The problem will be a buffer overflow. One question is whether it is the receive buffer or the send buffer. 

My best guess is the receive buffer: What could go wrong with your code is that the interrupts used by SoftwareSerial take precedence over the ones used by HardwareSerial. So maybe that's the reason why nothing is sent before the full message is received.

The SoftwareSerial library has a debug flag (_DEBUG) which you can enable to pulse a pin as an overflow indictator.

What I would try is to swap Software- and HardwareSerial (i.e. use SoftwareSerial for the communication with the PC). If the problem is as described above it should then vanish because the HardwareSerial does not block interrupts (nor the program flow) while receiving a byte.