LevelHeaded makes a good point: is yours a case of ulnar sublux or impingement? You should ascertain this information.
If it is impingement, the following information MAY be helpful.
1) As LevelHeaded and BBB suggest, you incorporate deep tissue work and wear wraps for support and warmth.
2) Movements which involve extension at the radial ulnar joint IN THE ABSENCE of movement at the shoulder will most likely be contraindicated. They might be okay for warming up with an ultra light resistance to warm up. I would personally set up a band or cable station so you can perform what looks like a straight right (boxing) and straight left. This will allow movement at the shoulder, allow the scaps to move freely, provide some anti-rotational work in a closed-chain manner.
3) Obviously, as the weight increases on the above exercise, it becomes primarily an anti-rotation movement. At this point, I would incorporate bw pushups at a moderate angle. The progress to regular push ups. Then, possibly feet elevated or weighted push ups.
For all variations of the push ups, I recommend using perfect push ups, blast straps, or something similar. Because the handles on the perfect push up, blast straps, etc. allow for subtle rotation (do NOT do that drastic twist in which the upper arms go to 90 degrees to torso).
Diamonds will most likely be a contraindicated movement for you.
Before the rectus abdominis, internal/external obliques, transverse ab fatigue to the point the erectors, quadratus lumborum, multifidus, iliopsoas, etc. synergistically dominate core stability, move to #4
4) DB presses on a low incline. I suspect dbs will not lock you into the internal rotation a straight bar will. Yes, yes, yes...I know one can bring the elbows into the safer 45 degree position with a barbell. However, the dbs should allow you to find a pattern that better stimulates the three heads of the triceps WITHOUT irritating the impingement (if it is an impingement). For added anti-rotational work, one arm db incline presses are a good option (they also allow you to self spot if necessary).
You can even reverse step 3 and 4 as a form of pre-exhaust. This method favors those with less than optimal anti-extension strength in the core.
Most people are not strong enough to perform the sternum pull up. You can use the lat pull down with floating handles, lean back so the torso is not perpendicular (ie instead of the head pointing towards 12, lean back so that, with a neutral cervical spine, the head points at about 10). This will sufficiently recruit the long head of the triceps because it originates at the scap and will therefore assist with adduction in the sagittal plane.
If you noticed a common theme, you're right. For you, tricep training will most likely involve exercising them in a synergistic role of pressing and (some) pulling in which the hands start above the torso (to activate the long head).
You also entertain the possibility that straight bar pressing/pulling are contraindicated.
And, of course, use pain as the final arbiter. In other words, if it hurts, stop immediately and re-evaluate.
To see vids of the floating handle concept, I have two vids in this thread:
(in one of the vids, I attached to a pull up bar but it just as easily be done to a lat pull down or hammer strength to allow proper float.