First, this architecture won't work because you can't present a Network Firewall endpoint publicly. You must have something that has an Elastic IP attached to it (so in this case, probably Application Load Balancer) and then direct the traffic to Network Firewall.
Second, I wouldn't recommend this architecture. Given the features available in APi Gateway (in terms of pulling inbound requests apart; checking them for validity; and acting as an ingress point for all requests) and that you can attach WAF to API Gateway I would question the value of adding a firewall to the mix. In short, I think there are better ways of doing this although I totally understand that customers are often comfortable with the approach of "but this is the way we've always done it".
If it were me, I'd use API Gateway and (maybe) WAF to go with it - because it provides a high level of security for a low level of complexity.
Given the complexity of this design and the customer requirements I'd strongly recommend that you get in touch with your local AWS Solutions Architect to discuss the best way forward. Not trying to avoid a conversation here - more that this is something much easier accomplished interactively rather than in this particular format.
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