I would certainly continue to install this setup without altering the cabinet.
The main reason is that ABS is cheap and easy to glue for dollars, definitely try this first to see how it fares and then you will see if you need to gut the wall out to lower the drain inlet.
You're going to try to cut that black tailpiece as short as possible, double trap each one, then elbow your way to a tee to the drain in the wall.
The only problem I see, and only if it is a problem, is that the trapped water will be rather high, but it looks like those white parts is also part of the tailpiece too, and your stoppers go above them.
So I would certainly try to spend $15 and 30 minutes to try this setup before tearing out a whole lot of stuff that can open a lot more problems for a DIY.