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According to the data provided by fellow members and reading my own books the Fw 187 very well could do that.
1940 is pretty irrelevant because the Luftwaffe didn't want a heavy fighter,as defined by various contributors to this thread,at that time. If the organisation that orders and pays for aircraft doesn't want one like the one you are hoping to produce then you won't sell it to them.
That seems a more relevant question than hypothetical discussions about what it could or couldn't have done.
Sorry if I may have been repeating the points that other members already posted...
RLM refused to let Focke Wulf purchase DB601 engines. That was the kiss of death for the Fw-187. They killed the He-112B and He-100 using similiar bureaucratic methods.Tank then took the design directly to Wolfram von Richthofen, chief of the development section of the Technischen Amt, the research and development arm of the RLM. Richtofen was not so convinced that bomber performance would remain superior to fighters, and gave the go-ahead for the construction of three prototypes, but on the condition that they replace the DB 600, which was in extremely short supply, with the less-powerful 515 kW (700 PS) Junkers Jumo 210.
Well I know they DID take a few Me 109 Ks with them, so there's that. Naturally they were more interested in new technology, such as jets.