I was wondering how I might mitigate the problem even little bit. Then it occurred to me. I can put a tee in the line and run a tube vertically. That'll give the air a way to escape. And if the tube is tall enough, I don't have to worry about the water overflowing out of it. It just needs to be higher than my highest rain barrel.
So that's what I went about doing. Initially I used a section of irrigation tubing. Then it dawned on me. If I use clear tubing I'll be able to see how high the water is in the rain barrel (water always seeks its own level). This solves problem #2.
As luck would have it, the second bed along the drip line has a trellis. So I zip tied the tubing to it. You can hardly tell that it's there.
This picture shows the water level with the barrel turned off. As you would expect, the water is below the level of the emitters.
I did see a few bubbles escape up the tube, but I still have some emitters that don't appear to drip much. Air could've already been in the line after the tee. Who knows... I'll just have to keep tinkering with it. I haven't had any plants fall over and die yet from being too dry. Maybe what I have is sufficient and I just worry too much.