How should I encode my 4K video stream?
As mentioned by @Mulvya, I think the issue is the "slow" preset. 

One option is encoding a compressed stream that is also lossless. For example, the x264 encoder can do this using "-preset ultrafast -qp 0". This will probably result in failing your 3rd requirement (compatibility), but since it's lossless, you can transcode later onto cheap spinning disks. You can also look into using the huffyuv lossless encoder, which might be better supported. There are some gotchas with this method, so you definitely want to test your workflow from beginning to end. In particular, see this question:
Using h264 in loseless mode brings small unexpected results

If you do go this route, I would also experiment with dropping subsampling and encode at 4:4:4. It may not increase your file size by that much and will help you if you plan on doing any post-production work.

Another option, depending on your budget, is purchasing an encoder. I love FFmpeg, but it's meant to work everywhere, not to take full advantage of your hardware. Mainconcept isn't too expensive and will better be able to use that i7 processor you already shelled out for. (See https://software.intel.com/en-us/articles/using-the-intel-media-sdk-within-mainconcept-h264avc-encoder-for-intel-quick-sync-video for example.) They also offer an HEVC encoder which would further shrink your file size. I can't vouch for the quality, however, as my experience is with Ateme. But I believe you can at least evaluate it out for free.