Adding a dog wash sink in my basement

I have an open area in our utility room that would be perfect for a dog wash sink. I've found a prefab one at lowe's that seems nice enough. Comes with a faucet and drain with maybe 5-10' of flex pipe and looks like an integrated trap. I think it's also designed to help with dog hair specifically for easy cleanout.

So if I buy this sink, I would need to tap into the existing water lines nearby which I'm fine with. I've been doing a bunch of plumbing work the last year so I feel pretty comforatable. The W/D is right there so I plan to take the lines that feed the W/D and tap into them with sharkbite tees and then use PEX to bring it around a corner up against the wall to the sink.

On the sanitary side, I will use the flex hose and run it back around that same corner and then tap into the existing sanitary line that fees both the washer as well as a sink/toilet on the other side of the W/D wall.

The walls are all open so that makes things relatively easy for access. What I need to know is what trouble am I looking at tapping into the sanitary? I know to keep everything at a 1/4" slope when moving horizontally. From there my plan is to tap into the existing sanitary either vertically with a sanitary tee at a vertical drop or horizontally with a wye fitting. Just depends on when I actually measure everything out what I will have space for. Do I need to add an additional vent? I'm in the basement so getting a stack up through the upper level and into the attic may be a non starter. That takes this project from something I can definitely handle (tapping into PVC/Copper) and turns it into a much larger construction job where I'd have to drill up through some bearing walls, go through my attic and then through my roof and patch the shingles/waterproofing/etc...

Can I get away with no vent on this new sink? Or is there some kind of vent I can use that vents into the same room? Obviously I don't want smells or odors inside the house but if it's just a way for air upstream of the p trap to leave the system that shouldn't be an issue right?