I'd encourage caution. The BMIP 4K is known to have severe (as in not-working) issues with OBS Classic at least, and may still have them with Studio (I haven't heard anything from anyone in the last month or two). Not to mention that they released it with a fan profile that made it sound like a dustbuster any time the PC was powered on... though I think they put out a firmware update to fix THAT, at least.
You can do local recordings at the same resolution on Classic for almost no additional CPU overhead, just costs the disk iops to spool off a copy to the local hard drive. Make sure you record to FLV, MP4 can cause high CPU issues due to how the container format is structured, and is a terrible idea to record to for many, many reasons.
On OBS Studio, you can record locally at a higher res/framerate, and set the CPU preset to Ultrafast for the local HQ copy and follow the high quality local recordings guide to throw a mountain of bitrate at the problem (and bitrate is cheap when you're just writing to a local disk).
There IS a very minor performance hit, yes, as it has to encode twice. But it's usually less than 5% CPU even for 1080@60 on a modern i7. Plus, it's free compared to the cost of a cap card and associated cables and other hardware needed to make it work.
Livestreaming is very CPU intensive. You'd want to keep that on the i7, not the 'spare parts' machine. So you're going to have a pretty big hit already, and spend quite a bit of cash on a cap card and headache on getting the audio working properly (like getting your mic over to the recording machine with the cap card, and still using it on the gaming/streaming PC).
I'd strongly advise just copying to disk, or using Studio if you want a high-quality local recording. It will be far cheaper, and the CPU hit for local recording is next to negligible, especially compared to the MASSIVE CPU LOAD incurred by livestreaming. Which your 'spare parts' machine isn't going to be able to handle anywhere close to as well as the i7.
It's why I don't use my old i7-920 as a dedicated encoding machine, when I have a 5820k as my gaming/streaming box (and do archive broadcasts locally). It wouldn't be worth it at all. If you want to do a 2PC setup, the second system's CPU should be equal or STRONGER than your gaming machine. So an i5 gaming rig, with an i7 or beefy multi-CPU Xeon encoding box as the second system.
