Update (Version 1.7.0)
After 1.5 years
the chat feature is back!

The solution is completely my own creation and thus should not really need updates in the future as our host upgrades us to newer versions of PHP, etc.
If it does end up needing work after software updates, they should be very quick and easy to implement.
Right now these are the main things implemented:
1) Send and receiving messages (of course).
2) Chat window auto-scrolls to the bottom at page load and after new messages (if not scrolled up higher).
3) The chat message entry field is focused on page load.
4) Pressing [Enter] sends messages. (In addition to the "Send" button.)
5) Chat messages with URLs will auto-link the URL just like in forum posts.
6) Usernames are colored and linked properly.
7) Message timestamps can be displayed (hover the username for that message).
These are the new features I plan to add very soon (the first 3 all rely on similar new back-end features):
1) Chat room member display. (Who is currently "in" the room?)
2) System messages when a user enters/leaves the room.
3) Possibly some meaningful per-user rate-limiting.
4) Maybe display new-message-count in the title of the page if new messages arrive while the chat tab isn't currently visible/active.
5) Maybe play a chime sound if new messages arrive while the chat tab isn't currently visible/active.
Notes:
Special characters don't work, but they don't work in forum posts either. I may need to look into that when I have time.
Currently a max of 450 chat messages are displayed at once, and anything older is perma-deleted. This will be tweaked if needed for the best user experience.
Individual messages are limited to 1,500 characters or fewer. This seems plenty.