The solution I ended up with was suggested by the Dev and works without needing any Safari extensions. It seemed counterintuitive at first, but works perfectly without affecting right-click functionality for macOS in general. This process removes the "conflict" macOS has with trying to have two different functionalities with right-click with two-finger taps.
Turn off right-click in the macOS preferences then immediately replace that functionality with BTT. Then use BTT to enable Safari to open links with a 2 finger tap. Steps below.
Step 1 - Disable 'Secondary click' in macOS system systems for Trackpad:
Step 2 - In BTT, under Gestures/Triggers - Trackpads - Global - Make a '2 Finger Tap' gesture assigned 'Right Click':
Step 3 - In BTT, under Gestures/Triggers - Trackpads - Safari - Make a '2 Finger Tap' gesture assigned 'CMD+Click':
That's it.
You're now able to two-finger tap on links in Safari to open them within new tabs in the background. Of course, you need Safari enabled for it in the first place:
If you still want to see a contextual menu for Safari links, you can hold down the control key and tap on them.
Not sure why, but this BTT behavior doesn't work while I remote into a Mac though Apple Remote Desktop, but it does work when I'm using the machines directly, but that's the only glitch I've found.