I cannot remove my sink drain cover as it has four holes which engage the separate strainer I use to fill the sink. Baking soda will not pass the holes and drop into the drain.

I have the same sink drain cover. I use baking soda and vinegar first with hot water to deodorize. Depending on where your outside garden hose is located, you can try clearing your sink with more force directly. Make sure your garden hose is clean before bringing it inside your home, or lay a sheet or tarp down where the hose is going to come in. Then have someone hold the end of the hose pointed down into your drain with a rag around it. Have someone to turn the on water full blast and wait for you to call out "turn off," then repeat.