Create the initial raster from the Numpy array in Python using rasterio. It should be included with QGIS. If not, just install it.

See example at Rasterio - Creating data

1. load and reshape the Numpy array
2. create the raster with rasterio

This should be easier. Rasterio can also handle reprojection

This Page seems to have the answer you are looking for. You need to use rasterio.warp.calculate_default_transform()

1. Create your raster in EPSG:4326 as you have already done

2. reproject according to the method in the link (just tested it - it works)

    from rasterio.warp import calculate_default_transform, reproject, Resampling
   
    filename = 'my_4326_file.tif'
    with rasterio.open(filename) as src:
        transform, width, height = calculate_default_transform(
            src.crs, dst_crs, src.width, src.height, *src.bounds)
        kwargs = src.meta.copy()
        kwargs.update({
            'crs': dst_crs,
            'transform': transform,
            'width': width,
            'height': height
        })
        outfilename = '/Users/zoid/Desktop/my_32755_file.tif'
   
        with rasterio.open(outfilename, 'w', **kwargs) as dst:
            for i in range(1, src.count + 1):
                reproject(
                    source=rasterio.band(src, i),
                    destination=rasterio.band(dst, i),
                    src_transform=src.transform,
                    src_crs=src.crs,
                    dst_transform=transform,
                    dst_crs=dst_crs,
                    resampling=Resampling.lanczos)
   

This resulted in a transformed raster that matches the untransformed one.