TL;DR: Will compression hurt? No. But it probably won't help you if your goal is a very responsive game.
You have two problems to solve with connectivity...
Can I get enough data to the client in time?
In other words, is my bandwidth enough? Is the "pipe" I have to send data big enough to carry the data? If you can't send enough data in time, then compressing it helps as it lets you send more down that pipe in the same amount of time.
Can I get data to the client fast enough?
In other words, is my latency low enough? Is there enough "pressure" in the pipe I have high enough to push data quickly? If you can't get data fast enough, then compression doesn't help. If you can send all the data you want, but not fast enough (no matter how small the data packet), latency is important.
Bandwidth ("Can I get enough data to the client on time?") is where compression will help you. Think like streaming a high quality video - you need to send as much as possible as fast as possible. But unless you are watching a live event where seeing what happens immediately is important, you don't care if it's a few seconds behind.
Latency ("Can I get data to the client fast enough?") is where compression will not help you. It doesn't matter how much data you can send if you can't send it fast enough. Compression doesn't make it faster, it just means you can send more. You could want to send only 1 byte really fast, but, if the connection is slow (high latency), compression won't do a thing for you.
So based on your desire for a very responsive game, then no - compression won't help you. That's because you want very low latency. If a player presses a button to trigger an action, you want your message to get to the other player as fast as possible.
If you do think your latency is low but your data still lags, then yes maybe compression will help. But you're better off focusing on