Well, I installed OBS as Josh suggested and it has everything.
Speaking about small and portable, I have one of these that I sometimes carry with me. A fully functional computer (minus the monitor) and yes, you can play RA2 on it
Haha, that's a Raspberry PI 2. Are you serious about RA2? I assume you didn't try it. Sounds like a contender for a 5 FPS slideshow at lowest resolution, at best, to me.
My PC isn't all that much bigger though! It's just 10cm x 10cm (4" x 4") and 3.5cm (1.5") high.
Software can't really make that much of a difference in speed (except over-clocking software) imo anyway, and I've been through what you're going through now. At the end I went 'fk it' and bought a much better gpu. There are several decent gpu benchmark sites out there but I've found this one to be pretty good:
The choice of the right software does make a huge difference. So much more so than overclocking that you can't even compare them.
Your Intel HD 5500 doesn't rank high enough in my opinion to record in your reso, especially when there are a lot of units on the screen.
Video encoding isn't usually done on the graphics card. While it's possible, it's definitely not the norm. Your link is most probably about 3D performance and hence irrelevant.