First, I am only a DIY plumber and not an expert here.
Personally, and even though the actual drain is smaller, I would use 2" for the lav sink also. And, I believe you need vents behind the lav and between the tub and washer.
The biggest problem I see here, however, is that the basement bathroom is going to overflow if your septic system ever backs up ... and think of that this way:
If you only had the basement bathroom going out into your septic system, you would immediately know there was a problem if the tub drain began running slow or the toilet would not flush properly ... and now go upstairs and consider what is going to happen when you flush there and the septic system backs up.
Even though your slab-out drainage seems quite convenient, I would nevertheless pump that basement bath out through a sump and pump with a check valve.