So I've had this problem since Beta 6. It does seem to be an issue with Gatekeeper, but it very well may have started with VMWare Fusion 12 for me as well. Let's get crazy now. This only happens on my two 2019 Mac mini's. Not my 2019 MacBook Air. I use one Mini for work and need a Windows VM so I did install VMWare Fusion 12 to test that. I don't remember if that was when this all started or not. But I did install it on both Mini's. That may not be a coincidence. I have had some limited success, however. Here's what I do (So far about 75% success rate):
Wait for Gatekeeper to time out and just LET. IT. GO. (I usually go do my grocery shopping or watch a movie.)
Reboot the machine and log in with my admin account.
Open System Preferences >> Security & Privacy and unlock it immediately. (Leave it open. Once that PrefPane locks you'll need to find another movie to watch or you'll stress eat all your groceries.)
Re-run the installer or application that will request access to something.
At this point I've noticed something different. The dialog box I get still has the Open System Preference button, but the Cancel button has now been replaced with OK. I assume because the PrefPane is open and unlocked. I click OK and it gives the application the access it needs and I can verify in Security & Privacy that it is in fact there. Now my application runs as expected.
I haven't had time to do any in depth detective work. I will get to in the next day or so. I'll be sure to come back and let you all know what I find out.