Elbow bursitis

40 yr old athlete here. Recently resumed lifting weights after a few years doing other activities like cycling. During that period, I did pushups and other body weight exercises but no weight lifting.
I believe I developed elbow bursitis due to the way I sit at my computer desk, resting my upper body weight on either elbow. Now, I experience pain at the tip of my elbow during bench pressing and tricep extensions (lying on my back and extending triceps).
It took a few weeks for the pain to get to the point of preventing me from performing the exercises altogether.
My elbow is pain free except during exercising.
I can feel the bursa moving around on the tip of the elbow on one arm but not the healthy arm.
Touching the bursa does not cause pain, only those lifting exercises.
I have been icing the elbow several times a day and taking Aleve. After a week of self treatment, I tried working out but the pain was still there.
I still work out but avoid those exercises that cause elbow pain.
How long should I continue this self treatment before I see a sports medicine professional? Can I anticipate a pain free work out someday? or is my elbow bursitis a chronic injury here to stay? Thanks for any advice you may give.