As others have said, Pilates is focused on core strength. I reckon for most of the population that’s the most important factor in back pain but for cyclists you need to sort out your Hamstrings (legs) first. If you bend forward with straight legs how close are you to touching the ground?
I’ve only been to a few pilates classes but found them boring. I’ve found lots of yoga classes I enjoy.
Loads of variations to yoga. My recommendation –
Start with Iyengar – slow and precise. Can be a bit dull but teaches you the positions you’re trying to get to very well so you’re less likely to hurt yourself.
Once you’ve got the basic positions really sorted move to something faster – some form of Hatha/Ashtanga/Dynamic yoga (or do both sorts at once).
Try as many different teachers as you can – other than Iyengar (which is very prescriptive in how it’s taught) the rest is the wild west so some instructors have a ‘fitness’ approach whereas others are playing whale music, burning incense and chanting (which may or may not rock your boat).