Or perhaps lighten the weight and just work on form?
What is your current level and experience with push ups? Have you successfully bolted on other programs with S&S? Are you a heavier or leaner practitioner?
If you have a solid base, switch to a “keep the skills fresh mode”. Example: 50 swings, 4 GUs, with 24kg 2x per week. You can weave in the push ups and MONITOR recovery! You may also find that you need to have an inverse relationship with push up volume and S&S work.
If you don’t have a solid push up base, I suggest taking a couple months to hone the skill. Take the max number of push ups you can do (with good form of course), and then space that over several sessions throughout the day a few times a week first. Easy right? Very easy in fact. Give it a 10% bump the next week and the week after that. Week 4 deload. Test new max, repeat another 1-2 months. After that consider doing the challenge.
I am not the king of push ups, but it will allow you to gauge a couple things. Easing the movement in will give you an indicator how your body will respond. It will also gently get the CNS use to the movement and likely not impede recovery. It will also allow your connective tissue a chance to remodel according to the new stimulus.
Not trying to discourage you, but perhaps help you take a more measured approach that will serve you better in the long run.
