You can't wet vent the kitchen sink and the washer over the toilet.
Those need to come in below the toilet. And they need to have their own vents, which can tie into the vent stack above the flood level of the highest fixture.
You need to install a double fixture fitting, if you intend to use a cross fitting. Just for your own knowledge, when you flush a toilet with a sancross, the water skips over to the other side. That's why the inspector wants you to cap that side off.
Heck, with the new toilets I don't like using any kind of cross now. I prefer to bring them in so that the waste points its way down the stack, not allowing skip over.