|Audio/Visual equipment is a little out of my territory, but as a computer person, I understand your dilemma.|
In my opinion, purchasing hardware (Wired or Wireless) to display your computer screen on your TV would be the easiest solution. However you can set your computer up as a DLNA server which would allow you to stream pictures, video and music to any other DLNA compatible device on your network. According to Sony's website your Blu-Ray player is a compatible DLNA device. See the link below for instructions on this:
A third option may be connecting your External Seagate Hard Drive to your Blu-Ray player. It sounds like it will play the contents of a flash drive, maybe it will recognize the External Hard drive and allow you to play it?
Law of Logical Argument: Anything is possible if you don't know what you're talking about.