Reddit mentions of The Complete Book of Pilates for Men: The Lifetime Plan for Strength, Power & Peak Performance

with you having back problems and what suggests a weak core, take a look at Pilates. It can strengthen your core correctly and hopefully end the back pain when doing situps (your hip flexors are probably doing all the work). It could be used along side your weightlifting and improve your sets. SeanVigueFitness does pilates/core work videos you can do at home.. If you want a book, I suggest The Complete Book of Pilates for Men: The Lifetime Plan for Strength, Power & Peak Performance.

So what you're describing, doing pilates on BWF rest days, is what I initially set out to do. The problem with Pilates is that the better you get at it, the more taxing it becomes. Eventually I just couldn't recover successfully.

What I do now is 4 days a week of Pilates. I do squats first as a warm up for pilates, then do my pilates workout, incorporating some BWF movements in at the end. For example, the last movement of the Pilates traditional mat sequence is pushups, so that is easy to work in. I also do pullups at the end as well. Two days a week I finish off with wind sprints.

I don't do anything for core, because why would you need to? All in all it is working extremely well. I feel very strong, much more flexible, and don't have any of those nagging pains that I used to deal with.

If you really want to get going serious into pilates, I highly recommend the book linked below:

https://www.amazon.com/Complete-Book-Pilates-Men-Performance/dp/0060820772