@Realjas If you haven't already, please start with a clean uninstall. Otherwise, the hidden library files corresponding to Pets (or any other packs you once had installed) can get in the way of your launching the game going forward.
https://bluebellflora.com/how-to-do-a-clean-uninstall-and-reinstall-on-a-mac/
Then reinstall the base game only. Next (or start here if you already did a clean uninstall), run the Super Patch (digital download version) to patch the game to 1.67. The launcher doesn't always do a thorough job of patching on its own. I don't know where you found the update you listed, but this post has a link directly to EA servers:
https://bluebellflora.com/sims-3-manual-super-patch-for-mac-1-67/
Launch the patched base game to test. But leave Origin out of it; you can launch directly from the TS3 icon in Applications. (Mojave's security features block Origin in-game, which can crash the game you're playing, but there's no benefit to having Origin open while you play anyway.) If the base game runs, you can install Pets and any other packs you have. Four of them (Ambitions, Late Night, Fast Lane, and Outdoor Living) don't install correctly and need a couple of workarounds. The first is simple: create an alias for any pack that doesn't have one in Applications, alongside the alias for the base game.
https://bluebellflora.com/sims-3-packs-not-showing-as-installed/
The second workaround is to copy files from those four packs into the folder for your newest pack by the date EA released it. This video shows you exactly what to do:
https://answers.ea.com/t5/The-Sims-Mac/CURRENT-ISSUE-Sims-3-Unknown-Error-in-El-Capitan-Sierra-High/m-p/5725514#M5926
If you're not installing any of the four problem packs, you can skip both of those workarounds. But if anything goes wrong, please post back with a list of the packs you're trying to install, as well as any error messages you see.