If BeBr₂ had two double bonds, Be would be surrounded with more than eight electrons.

If BeBr₂ contained two double bonds, the sum of formal charges would not be equal to zero.

If BeBr₂ contained two double bonds, the formal charges of Be and Br atoms would be maximized instead of minimized as required by the formal charge rules.

If BeBr₂ contained two double bonds, both Be and Br will violate the octet rule.

If BeBr₂ contained two double bonds, each of the Cl atoms would be surrounded by more than eight electrons.