Basement Laundry Plumbing

I am redoing the laundry and plan to integrate a sink into a counter which will also go over the washer and dryer. I have the original drain (from a 1947 built home) and would prefer not to drop the washer discharge directly into the sink.

Can I plumb a discharge for the washer with a P-trap that connects to the drain of the sink without having additional venting beyond a “mushroom.” Subsequently I also have a humidifer pump drain to drop into the same drain, but this is only residual discharge.

I want to ensure there will be enough venting, and connecting to the vent stack is not an option, as I have a cleanout, toilet, shower, and upflush toilet discharge connected. It also doesn’t work because I’d like to add cabinets where I would have to connect.

Thanks-