Previewing something in obs that does not include osu does not cause lag. Moving over to a scene which includes previewing osu however triggers the issue. I also just found that the issue doesn't resolve itself after closing obs. I need to restart osu in order for the frame rate to recover.
I may have just found my answer to fix this problem though. I have been using game capture in order to record osu, however it seems like this issue does not occur with window/display capture. I have not tested this for an extended period of time, so I'll see how this form of recording osu fairs. If I do run into issues while using display capture I'll come back here.
I have tested this with all available versions (except beta) of osu. The only versions that this issue doesn't affect are Fallback, and any version with compatibility version on. Both Stable and cutting edge has this issue.