I would like to reproject a raster from WGS 84 (degrees) to ESRI 102001 (meters) while simultaneously resampling the original pixels to a coarser resolution (e.g. .00001 deg to 30 m). I want the final pixel values to be the weighted averages of all high-resolution pixels within the bounds of each final pixel. See the attached rough sketch for an idea of what I'm going for. I want an average of all the small pixels in the red outline (large pixel in different projection).
Concept image:
I have tried gdalwarp using the average resampling method but have found this to be inaccurate. I know I can reproject the original raster to ESRI 102001 at say 3 m resolution first using NN, then resample up to 30 m using the average method, but I would like to be retain as much accuracy as possible and do this all in one step. Is this possible?