Only #3 is natural. If the subject of "buy" is different, then use a that-clause. He insisted that we buy a car.
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ljswaveCan #1,2 be used and make sense totally?No. They are not correct.
ljswaveI have thought #1 make sense though not grammatical.It's almost pointless to rack your brain asking if an ungrammatical expression makes sense.
ljswaveIf the subject of 'buy' is differentJack insisted on buying a car. ~ Jack insist [ Jack buy a car ].