Plumbing – How to get the sink’s trap to fit around the rest of the drain assembly

My suggestion is to dig it up from outside and work through the pipe from there over under the basement floor. If it is a simple blockage you should be able to get it from there. It is a collapsed red clay time then you can measure accurately so you will know more accurately where you may have to cut into the basement floor. If the blockage is tree root growth into the drain the outside access will be much better for getting in a power chopper/rooter device.

Lastly when you do go to cover the dig back up consider adding an inline access to that part of the pipe for the future or re-access the TEE joint with some gradual turn joints.

Is there a joint under that escutcheon?

If so, why not run an elbow toward the back wall, then an elbow upward to a vertical pipe, then an elbow forward to a conventional p-trap?

This would effectively reduce the risk of siphoning water that an s-trap creates.

If there is not enough distance in back of the pipe, the whole elbow/p-trap thing could be canted to one side or the other.

One or more joints could be eliminated if you can find a run of pipe with an integrated elbow. The lower horizontal run is needed only if the elbows don't move the vertical pipe far enough back to accommodate the p-trap.