What you have to remember, is that volume in training, does not always equate to performance in the test. Its just not that easy
If your tendons are grumbling, then its a sign you are doing too much of something, and possible creating an imbalance in the body, so lets look at the various elements in a push up, and elimenate the irritating factors, whilst maintaining work that will help the exercise as a whole.
A push up is much more than a chest, shoulder, and tricep exercise. In the prone horizontal position, you are stressing your core, your upper back muscles. your quads, and even your calves. If any of these do not function in a stabilising fashion, your form goes south, and your push ups suffer. The upper back muscles also act as antagonists, and engage hard on the downward motion of the push up. The stronger they are, the less the chest has to contract negatively, and the more push ups you will be able to do.
So, in order to maintain a good level of push ups, without doing push ups, lets construct a training regime that will allow your shoulders to rest, but stress components of the movement.
1. Bent over Barbell Row. Overhand grip, weight through your heels, make sure you squeeze through the back and traps at the top of each rep
2. Plank. Concentrate on perfect position. Lock in the core, the quads and the ankles. Become as immovable as a face on Easter Island. Strong plank = perfect push ups.
3. Underhand, close grip pull ups. Get a full stretch at the bottom of the movement, then draw yourself up smoothly, pushing your hips forward, with tight abs, so your upper chest to the bar, not your chin. Pause at the top, and bottom, of the reps.
4. Tricep pushdowns. Keep the shoulder blades pulled back to 'open' your chest/shoulders. Abs tight, and go for moderate weight, and full technique. Do not allow yourself to 'muscle' the movement by hunching shoulders over the bar, or leaning too far forward.
Work on these exercises, in that order, 3 clean sets of each, 3 times per week. During this time do no chest work, no shoulder work, no push ups AT ALL.
Give it 4 weeks and see how you feel. I guarantee that you will not lose push up performance in this time.