Hi John, yes that product has a very high % of sodium hydroxide and needs to be used with extreme caution, all you can do now is to deep clean the black damaged paint as best you can let it dry and then use a heat resistant black paint probably best to apply two coats, if your customer is not happy with the results you may have to replace the door.
you should have no problems in sourcing an aerosol can of black heat resistant paint, just google heat resistant spray paint. Tadgh