Your e36 probably has the same temperature/viscosity chart in the OM as does mine, although you have the largest available engine.
BMW recommended quite heavy oils for warm weather, so I'd probably use a 10W-40 at a minimum.
BMW did not require synthetic oils when your car was new, either, and drain intervals by the maintenance minder were pretty long, as well.
BMW later recommended oils meeting LL-98 for these engines, which would include M1 0W-40 and GC, both high HTHS oils.
If you're going to run 5K OCIs, you really don't need synthetic.
The six has a generous oil capacity anyway, so even a dino or syn blend oil should be capable of more than 5K.
I'm liking Maxlife 10W-40 in my little four cylinder, and I'll UOA this run after about 4K. Not sure how relevant that is to your entirely different six, though.
I also really liked the M1 15W-50 I used in the car for 6K last summer.
A number of owners use Maxlife 10W-40 in their old BMWs, while some favor 15W-40 dual rated HDEOs, as well as oils like Rotella T6 5W-40, a synthetic dual rated HDEO.
If you don't mind paying more, M1 0W-40 and GC meet BMW's current requirements, while M1 HM 10W-40 would also be a good choice.
Bottom line is that I don't think that any of the oils I and others have mentioned will hurt this engine in the kind of use you're giving it on the short drains you're running.
While you don't mention it, old German cars love to leak oil and to consume it.
Does your car either leak oil or have excessive consumption?
A thicker oil than what you're using would greatly reduce both.