This smells like nonsense... perhaps a grain of truth, but with a fairly hefty stretch.
(caveat, I'm a user, not a dealer)
I'm using a whole house of Unifi hardware - rotuer, switches, APs, cameras etc - all works beautifully with C4. No issues were ever so much as mentioned by my dealer.
The camera integration works very well - it is a $150 driver third party driver however. Snapshots of the cameras in Navigator don't display so you won't be able to get a video wall sort of display on your touchscreens or in the C4 App. But in the case of Unifi cameras you would be using the Unifi app to directly view/scrub streams anyway, as it's leaps better than using the C4 app to do so. You can get live streams, trigger events, and use zones on the camera stream as virtual buttons as well.
That said, C4/Snap would not "support" Unifi, being entirely separate vendors and all. If some sort of connectivity issue were to arise, and if your dealer were managing the network it would be them on the hook for resolving the issue - in only using kit from C4s list, the onus is more on C4 to resolve issues, and not your dealer. It could also simply be a case of them not wanting to install and manage Unifi in preference of networking gear they've standardised on, which they're certainly within their rights to do as well.
If I were cynical I could also say that there's a meatier margin to be had on the other gear they're suggesting. But I'm not. So I won't.
EDIT: joking, sorta. This is almost unquestionably the case, but there are other perfectly valid reasons above they would prefer to keep it in-house. Given the scale of the job I don't think getting another few hundred bucks on the router is really top of mind here.
In any case, AC-Pros are old hat. For Unifi APs, hardware from the U6 range would be a better choice.