So, I succeeded in convincing a governor to secede, but I was rapidly invaded and despite being able to hold off the invasion (I was flying a Stonewall-class battleship, which seems to me to be the most powerful warship in the game), I eventually lost my empire to declining loyalty. I was curious if anyone knows how to avoid this, and whether this means I can't start another?

Hmmm, I hope I can since I think I overwrote the save.

When you say play around with tax and production, I assume low tax high infrastructure is the way to go? How much influence does that setting have?

Basically or at least not the highest tax level. Believe generating a profit is often beneficial as well assuming you can manage it with a particular system. I know I had several smaller systems that the best I could do was break even. I haven't tracked influence that closely, but I've had systems in the 20s & dropping changed to over 50% & climbing. Of course in some I have the highest tax rate set, a net profit, but very high level of services & their loyalty is 90+ in those systems. I don't remember if I tried switching to that when their loyalty was low though.

Alright, may need to restart that game. We'll see. That was basically what I figured but I was trying to build defenses because despite me taking down a battleship or more a week the war wasn't progressing. Then again, as I realised when I did swing the war, it's not like you have anything better to do when you're ruling your own empire from the seat of your favourite battleship!

Update: I refounded my Empire on the back of another well-paid Governor (who I am currently married to), and follow a simple rule on most of my colonies: High Tax, High infra, Medium Production, Low Defense until 90%+ loyalty, and then drop infra to medium and start racking up surplus.