Try reenrolling in the Apple Beta Software Program. https://beta.apple.com/sp/betaprogram/redemption#macos
Download the "macOS Public Beta Access Utility" and install it. I had to do this the other day. You will see the macOS 10.15Catalina beta at the top. If you look down toward the bottom of the Software Update window it will say that another beta is available. Click on "more info". The macOS 10.14.6 Public Beta 3 should show up. It is version (18G59b). It is 2.97 GB in size.
That's the ticked. I search for that .dmg with beta access utility, but got that loop I was describing.
That link worked and I've got 10.4.6 Beta downloading now.