The wiring harness connector is the same physical part but they are not wired identically. There are three additional signals connected to the SRX cluster that are not present on the STS. This doesn't necessarily mean it won't work but you also have to consider the programming. I don't know if things like lane departure warning was available on the SRX so if you have it on your STS, it may not work (as an example). There may also be some differences in the messages displayed on the DIC. It could be difficult to know for sure if it's 100% compatible.
Automobile(s): 08 STS4 V8 F55 1SG Euro tails & 01 Monte SS Pace Car
No point to send it out to be programmed if it doesn't work. I would try to connect it and see what happened before sending it anywhere.
You might just be able to swap the display unit only rather than the whole cluster.