Forgive me if you gave us this info and I missed it, but what threading is the frame?
If English/ISO or Swiss, then the right-side cup has a left-hand thread, which is good because precession tends to tighten the cup in use. Then the wrench you use becomes less critical.

If the frame is French or Italian thread, the the fixed cup must be installed very tight or it will loosen in use. A normal flat wrench is marginal for that even with a steel cup, and asking for trouble with an aluminum cup.

A flat wrench with a Stein tool to hold it tightly against the frame is much better, but the right tool to use is the kind that threads onto the cup from the backside, like a Var or Campagnolo. Here's a Var #30, which is excellent because it comes with a variety of sizes of the jaws for different brands of cups that have different dimensions across the wrench flats.



That's a shop tool, too expensive for the home mechanic who'll use it too rarely to pay for itself. So I'd strongly recommend finding a bike shop that has this, or equivalent in some other brand.