Rule of thumb is what ever you do to one side of a vehicles axle you do the same to the other. You really need to get the other tire changed so there are 2 new ones of the same tread wear on the same axle.
You wouldn't change only one brake rotor or do the LF brakes and not the RF brakes. You do them as a set.
Not only will your transmission and differentials suffer excess wear due to unever wheel speeds your braking and ABS will suffer too.
If you had a 1970s vehicle sure you can get away with this and probably won't have any issues. But with a newer vehicle your only asking for more expensive issues.
Since it sounds like front engages less (being an open diff, I guess they use the brakes to bias torque between sides), I'd go there.
In my opinion, the concern over tire replacement is overblown, especially with systems like Audi uses. Subaru uses a full-time system, that would be more concerning, but even there I'm not so sure any more.
I'm sure your LSD is some kind of hydrostatic, that was Audi's claim to fame from the start. Hydroststics don't care one whit about different wheel speeds.
Even a Quaife (torque-biasing using gears) wouldn't really care all that much, and it would effectively be engaging all the time... Which they kind of are anyway.
Clutch-based LSDs (old style, just springs and plates) would be a concern because it would cause more wear there, but you don't see those on road cars. Modern clutch-based diffs like Bendix supplies to Honda (and other companies) wouldn't care, since each rear wheel is independently (wet) clutched, and the clutches are electro-hydraulically controlled. It would simply compensate for slip, like it's already doing all the time anyway.