Apologies but more description is creating more ambiguity.
If possible pl paste your configuration and if possible you may show the port numbers in diagram which will help to troubleshoot.
However, since you are passing the same vlan across the firewall, Core switch and access switch, i do not see any need to routing.
However, you may want to check the Firewall rulebase permissions from Inside zone to DMZ zone and vice e versa.
Also keep in mind that, since you are using the same segement across the deivces you may need to pass the vlan 10 in all the uplinks correctly.