Zinc plated is vague -- means electroplated?
Zinc electroplating is usually prohibited for high strength steels (TS > 150 ksi) due to danger of hydrogen embrittlement. Medium strength steels (100-150 ksi) may require pre-plating stress relief & post-plating hydrogen bake-out. Zinc electroplating should be per ASTM F1941.
ASTM B695 is a spec. for mechanically plated zinc. It also has stress relief requirements for steels of TS > 1000 MPa.
Hot-dip galvanizing may be prohibited, as by your customer, due to hydrogen embrittlement in the acid pickling process, or thermal stresses when immersing into the molten zinc, but most probably, because the high temperature may soften the steel (additional tempering). Hot-dip galvanizing of fasteners may only be done in accordance with ASTM A153.
I cannot say whether your existing, plated SA-320 L7 bolts can be used. I see no language in ASTM A320 regarding plating, you must check the ASME SA320 spec. To convince your customer, do the A320 mechanical tests (Charpy impact at -101 oC & tensile test) plus hydrogen embrittlement testing for coated fasteners per ASTM F606.