On the remote Mac....
Stick with Safari and make sure that the Safari popup blocker is turned off.
Then visit Java.com and click on the "Do I have Java" button. This will ensure that Java is functioning within Safari.
On the Pulse Secure appliance...
Make sure that you are defining the process check on the proper tab of the HC Policy. There are separate tabs for Windows, Mac, Linux, etc. If you define the process on the Windows tab, it won't check a Mac.
If this doesn't do the trick, then make sure that Java is being permitted for your VPN URL within the Safari security settings. Safari > Preferences > Security > Internet Plugins (Checked) > PlugIn Settings > Java.
In most cases just "Allow" should work under the PlugIn Settings, but some things like the Premier Java RDP app need "Allow" and "Run in Unsafe Mode".