I'm more productive because I use some browser add-ons, and some other things that make producing some reports and doing a few other tasks much more straight forward
Well you've got some suggestions for specialist stuff from other folks, but I can't help but wonder if your problem wouldn't be best solved by installing said productivity enhancers on the computers that need them... Feels like you're jumping through hoops because - I assume - company tech support won't let you install the tools you need. In which case I sympathise greatly
Re capture cards... to my understanding they're meant for sending a video feed into your pc, e.g. for processing work, with no bi-directional interactivity native to them, so you'll have to have a hardware/software switch to route the mouse/keyboard input to the windowed machine. Which might well work, but may also prove more cumbersome than just using Synergy as intended...
If putting your personal pc on the company network is a no-go, would it be possible to give it a 2nd network card, or usb wifi dongle, and have a private network between the two? Oh, but that doesn't help if you can't install things on the company pc...
Boy, you sure do have a conundrum