How to use automator services in Fluid-created apps in Mojave
I figured out a work-around for part of my problem.  I'm not going to select this (my own) answer because it doesn't solve the entire problem.  The main reason I use Fluid to wrap galaxy in an "app" is because you can only have 1 galaxy browser window open.  If I have 2 open, changes made in 1 window affect the other, so when I'm working on 2 different projects concurrently, I was using separate Fluid apps to keep the projects from stepping on one another's toes.  I knew that private browser windows could also accomplish this, but I liked the convenience of a dock icon.

So I just worked out an Applescript, saved as an app, that opens a new private Safari window to galaxy.princeton.edu.  I then added that app to Security &amp; Privacy -> Privacy -> Accessibility.  This script is slightly simplified from what I actually used:

tell application "Safari" to activate

tell application "System Events"
    keystroke "n" using {shift down, command down}
    delay 0.5
    keystroke "http://galaxy"
    key code 47
    keystroke "princeton"
    key code 47
    keystroke "edu/"
    keystroke return
end tell


Note, depending on your accessibility preferences, Applescript will not type numbers, dots, or a few other characters, thus the key code 47 for the dots.

Now, whenever I run this "app" (which I gave the galaxy icon), it will open a new Safari browser window to galaxy.

This allows me to run my automator services on DNA strings in galaxy because Safari can be given permission to send apple events to System Events.app, whereas Fluid apps can't (I believe because they're not code signed).