You mention that for Full HD screen height usage is only 83%, with an AR error of 7%.
An optional solution that would work for most games is as follows:
Cut off 8 lines (e.g. 4 at the top and 4 at the bottom). Then you can scale one step larger, using 100% of the height and reducing the error to less than 2.8 percent.
The same can be done at 4K, with the same result.
I hope I explained it well enough. If not just let me know.