Ok, I got impatient and dug through the UT Tweak tool. Here's a workaround if you're comfortable in a terminal (or, in my case, adb shell):
gsettings get com.canonical.qtmir lifecycle-exempt-appids
For me, this had a value:
['com.ubuntu.music', 'com.ubuntu.terminal']
Then I updated this value:
gsettings set com.canonical.qtmir lifecycle-exempt-appids "['com.ubuntu.music', 'com.ubuntu.terminal', 'morph-browser']"
Voila! My browser is happily running in the background.