I had tennis elbow for about a year and mine was the same as yours: bicep curls hurt it the most, followed by barbell rows. What helped was work to strengthen forearms (PT) and grip, followed by as many if not more exercises for triceps than bis. The point is to have balance at the joint and in opposing muscle groups. Now, whenever I do bicep exercises, I always precede them and insert a tricep exercise in between each, the best ones being band work taught to me at PT that mimic work done for triceps with cables at the gym.
I found that I could often do compound weight work and lower body weights work, but had to avoid biceps work assiduously. However, even when I did compound weight work, it was not more than twice per week. I did loads of cardio during that time. I did not find that things like mountain climbers and burpees hurt at all, nor planks, so body weight exercises were OK for me. You'll have to experiment and see what you can do and what just introduces discomfort and pain.
It seems that we all experience 'tennis elbow' just a little bit differently, with the pain coming out of different muscle/ligament/tendon insertion points. One person's experience may not be yours. See what you can do and what you can't and make up a program for yourself for a few months based on cardio and whatever else your elbow joint allows. It's only for a short time and you will get past this. I actually ended up just taking most of the year off eventually, with just daily long walks (some other stuff going on in my life at the time) and it's OK< once you get back to weights, the strength will come back.
Clare