When using private IPs, the secondary (standby) unit never has internet access until failover occurs. Therefore, this issue seems to be related to the standby unit not having internet and/or not reaching the gateway.
That's likely the entire issue.
Which is why we don't recommend using that style of configuration on a primary WAN. For a non-default/secondary WAN it can be OK, or for internal interfaces, but both units need to have functioning Internet access, or at least functioning DNS.
Now if your private IP addresses on WAN can get out (upstream does NAT, for example), and your NAT rules on WAN are OK, then it's possible the units themselves could get out and be OK. If traffic leaving the firewall must use the CARP VIP to exit, then probably not.
You might try spinning up a local DNS server off the firewalls and then point DNS on the firewalls to that, see if it helps.