But HIIT is not automaticaly high impact. Why not use breathing squats? swimming? cycling?, barbell complexes?

Also, the volume involved in HIIT should mean a quick turnaround on recovery times.

If your joints are sore from a sprint drill session, then rest! Phase it so that your upperbody lifts follow over the next 48 hours, or work on your mobility/core stability instead - dont go jogging and doing empty cardio that will only serve to hamper your adaption to the HIIT
Don't let the door hit ya' where the good lord split ya'