I think you could shove directly as you are comitted anyway with 1/2 bet (you need to call $0.9 to play for $4.47 pot in case of a raise -> 20% EQ needed). But to be sure I made a little calculation.
EV (cbet) = FEQ*pot + (1-FEQ)*(EQ*final pot - investment)
BB's coldcalling range: (64 hands) QQ-88,AJs+,KQs,AJo+
BB's calling range vs cbet: (40 hands) QQ-88,AsKs,AsQs,KsQs,AsJs,AdKs,AhKs,AsKd,AsKh,AsKc,AcKs,AdQs,AhQs,AsQd,AsQh,AsQc,AcQs,AdJs,AhJs,AsJd,AsJh,AsJc,AcJs
FEQ (stands for fold equity) = (64-40)/64=0.375
pot = $1.55 * 0.95 = $1.48 //*0.95 stands for rake
EQ (stands for equity vs calling range vs cbet) = 0.2895
final pot (considering you shove as a cbet) = ( 2*$1.6 + $1.55 ) * 0.95 = $4.47
investment = $1.6
EV (cbet) = 0.375*$1.48 + (1-0.375)*(0.2895*$4.47 - $1.6) = $0.3638
His coldcalling range could be wider but his calling cbet range not that much. Cbet here should be fine.