Brownsea is owned by the National Trust. Try asking them about cycling on the island. FInd out if the ferry company accept bikes too. I was thinking it would be a no, but the NT are very pro cycling these days. You could argue that getting to the ferry by bike was eco-friendly etc. On the other hand, you could probably ride round Brownsea in 20 minutes, so is it worth it?
I recommend the Isle of Purbeck (which is actually a peninsular). Not that far from Sandbanks/Studland. Ride the two high ridges starting from Corfe Castle up to Kingston, along to Durlstone then over to Ullwell and up along back to Corfe. There are a few lung buster climbs and some awesome views.