However, I’ve read that tai chi takes a long time to learn and I only have a year to study with an instructor. Would this be enough time to see tai chi’s influence (and benefits) in my boxing or would I be better served cross-training in another art?
I think this will be enough time depending on the instructor. You won't be able to fight with Tai Chi but there are some principles about relaxing, sensing, rooting, and balance that you'll pick up long before you can start fighting with Tai Chi.
The most important thing is probably the following
1. Find and instructor that knows how to use and train Tai Chi as a fighting system.
2. Push hands has to be part of your training
3. Daily practice is a must.
The kung fu organization that I'm a part of train Tai Chi to help balance all of the hard hitting and blocking that we do. It helps to develop our soft and internal methods. Can I fight using Tai Chi, only with the parts that are similar to how I fight, so in short... Nope.
Have I used Tai Chi concepts? Yes.
Did they work? Yes, and better than I expected.
Tai Chi help me to recognize the small shift in my opponent. For example, if you punch me 3 or 4 consecutive times then I'll be able to tell if your stance is week or strong and where it's weak. This is important to me because I like to sweep.
It also helped me train to make my punches and movement begin and end at that same time. Physically it made a lot of my smaller muscles stronger. I still practice it today, but not as much. People who just do Tai Chi for health purposes aren't going to have a big affect on your boxing because their focus in tai chi isn't fo fighting. So things like punching and body movement are things that they can't teach. So make sure you learn it from someone who trains the fighting applications.
The fitness Tai Chi also won't be able to show how to drive power strikes. I studied Yang Style. The stuff that will help you will won't be the Tai Chi techniques, it'll be the Tai chi principles and methods that are used.