I think this as you said is your best option
Console -> OSSC -> HDMI Splitter ├-> Gaming monitor └-> Capture device
That way the only thing adding lag is your monitor and as mentioned, the OSSC does clean up the signal some so splitting the OSSC output will mean better quality on the monitor and better quality on the stream. Plus once the OSSC is setup, it makes setting up the stream easier. The Sync Strike keeps the signal 720×240 resulting in a flat image which you have to manually stretch in OBS.
Good luck with your setup and speed runs.