Wow! Thank you all for your replies.
I appreciate it very much.
Fortunately, I'm able to definitively say that the issue is not the Arduino because I can run a program on the Arduinos and have them communicate effectively at these high baud rates by jumper cable. It's only when I attach the HC-05's that the system fails, even with the same code.
However, if I drop the baud rates across the board (Arduinos and HC05's) to 115200, everything works as intended.
The other important thing of note here is that I'm not getting errors, I'm just not receiving any data at all. Nothing is being received.
@MartynC: I'm working with an already established system that requires a max speed of 375k because it reaches that speed at peaks. I'm just trying to develop a proof of concept of prove that I can in fact manage that speed.
Thank you all again for your help. Does anyone have other ideas what may be the issue because I am still facing this issue.