Try it without a screen first.
That way, if it works with low noise and good bandwidth, you save leakage inductance.
That layer of copper, since it is not part of the actual transformer windings, increases leakage.
If you look at a xfmr window, anywhere you see air or daylight, that is leakage.
Try a simple pri-sec test winding, like the Shure SM 57.
But first, see how many turns of the small wire you can get on the bobbin.
Then subtract a few turns, divide by your desired truns ratio, anbd then wind the pri fisrt.
If you have no room left for all the sec turns, no big deal.
So what if the ratio is 1:99, or whatever.