So my presses last night went pretty much the same. I had an old raggedy sleeve I used on that elbow to see if it would make a difference. When the snapping started to occur during the first working set, I put it on for the second and while it didn't stop the snapping, it made it much, much more comfortable and less distracting, like it was barely there. I then for my third set tried a wrap (I have for my knees in case that's ever an issue). I wrapped it tightly and it made no difference over the sleeve.
So I guess my final question to you is, if this doesn't actually go completely away this time, and it stays at such a mild level, am I going to be damaging the tendon by doing this over a long period of time?
I would think the human body would be somewhat able to cope with something mildly out of place like this, or at least adapt to it in a productive way. It doesn't hurt and right now isn't even uncomfortable with the sleeves.