CVG would get my vote. However, unless you are below and very close to footings, compaction of the sand hardly seem necessary to me. I'd just walk on the backfill and stomp it down by foot or you may squash the pipe with heavy compaction. The main need is to keep good support for the nearby footings.
On that score, check the pipe before you load it up at the lumber yard. When this stuff sits in the sun it gets brittle. Just step on it and see what it does. If it breaks, go elsewhere.
As to how thick a layer you need for using concrete sand as a filter, I have never seen where too little, other than none, is the case. In my graduate work when I had a research project to see how sub-drains do under highways (1954-6) I would dig into some of the backfill where water has brought along silt and clay and there was a sharp demarcation line at the junction of sand and clay, with no noticeable clay in the immediate voids of the sand. On that basis, I'd have to say "Use at least some concrete sand". Practically a few inches should take care of uneven excavation, etc. Amazing how this system can be done without being too fussy about it.
Using Ron's liking of fabric, at the tub, can you dig down to those holes, line the tub with fabric on the outside and then use the concrete sand as backfill?
If you feel the tub always draining nearby soil is of no use or a continuing problem, before you cover up your job with concrete, do some trials and see if you are causing problems by sealing the holes. Remember that you may have to use a few layers of fabric if what fabric you have may rupture from the forces on it at the hole.
As to the size of your trench, 2 ft. wide seems like a lot wider than I would do it, as long as I could get the drain pipe in proper location with suitable thickness of concrete sand around it. Again, I'd not close off things until I was sure what I did was working. You may have to add some drain lines farther inside the basement, since it sounds like you may have some water by-passing underneath the perimeter drain area and rising up farther inside.
Keep in mind, the best drainage systems cut off the water before it gets to cause problems, rather than trying to "draw-down" the water from a downstream position.
While you are at it, before you leave it as "suitable" drill some holes in the floor and see where the water table is. Do this before you do much of your digging and pipe laying, so you can judge how well your job is working. Seeing those results may help you to plan on any secondary lines.
Finally in your digging, remember to be careful of not undermining the footings. You might decide to partly backfill the job as you go along and them return and lay the pipe, taking only a short time to dig down, lay the pipe and then re fill. It would not be out of the question to ram a mix of concrete sand and gravel (a typical mix of concrete, but no cement)sideways under any footing appearing to have lost some soil support. The usual mix of concrete is 1-2-4. Cement-sand-Gravel. That means twice the amount of gravel than sand.
I hate to allow any gravel on these types of jobs or well meaning workmen will use that instead of the concrete sand. It shows how well indoctrinated people can be on an error prone procedure.