The BMW have had the 801E developed and produced in 80 copies. In July and August of 1942, the engine passed several times the 100 hour test. Due to redesigned pistons, heads, oil system and compressor, the boost pressure for take off and emergency rating (Start Notleistung) was now 1.62 ata, vs. 1.42 ata for the fully rated 801D - meaning 2000 PS for take off and 1710 PS at 5650 m (power values probably without the power needed for the coling fan - meaning 70-50 PS less power for the prop at these ratings). However, doe to the need to introduce too many changes in production lines, the engine was abandoned, later some features (mostly the improved S/C) were used on the 801S of late 1944 (probably the 'power section' of the 801D, used for the 801S, proved as capable to withstand the stress).
So yes - the RLM/LW/BMW/Fw could have had the better 801 in service in, say, second half of 1943.