I honestly don't believe a muscle imbalance in your back would contribute to elbow pain when doing pressing movements. I went through a phase where I was hitting the gym immediately after baseball practice where I was throwing 100s of times a day which inflamed my elbow tendons. This is what helped me.
Flossing. You can purchase a Voodoo floss band from Rogue, and honestly it's a great inexpensive investment. Prior to pressing, I want you to wrap your elbow one at a time with the floss band super tight. The pain barrier should be about an 8/10, and you will soon lose most of your blood flow to the area. Keep this band around the area for 2-3 minutes, up to 5, while forcing your elbow into flexion and extension (think hammer curl and tricep pushdown). Don't be alarmed if you become super vascular and your skin pigment turns gray-ish. Unwrap the band and immediately the area will become a flush red color as blood flow returns to the elbow/forearm. This brought my elbow pain from an 8-9/10 to a solid 1-2/10 or no pain at all.
Band pushdowns. Every workout, or at least 3x a week, I want you to wrap a mini-band (or monster mini, etc depending on your strength) around the top of a power rack and perform 100 straight reps of pushdowns. If you can't complete 100, rest pause until you force those 100 out (ex. 50, rest, 30, rest, 20). This will flood your elbow and tricep with blood and nutrients while also giving you a ridiculous pump.
By doing the above, I went from being unable to perform any tricep isolation other than extremely light pushdowns and kickbacks without pain, to doing heavy sets of skullcrushers and overhead extensions after heavy pressing days weekly.
I want to add that "maybe" using a knee wrap wrapped tight enough would work as a replacement for the floss band if needed. I've never tried it so I can't guarantee it. Either way a floss band is a great investment.