Belly fat is unfortunately just calorie reduction. You can’t target fat on specific parts of the body, just overall.

Improving core strength with no impact on the spine is tough, as most core exercises will likely engage the back and spine. I would try to implement standard, good form push ups if you can. Planking is also an option, but again form is king. Do them properly, for less time instead of “cheating” with loose form if you want to make improvements.