It is debatable whether getting a fellowship on H1B visa is any more difficult than on J1. While it is widely acknowledged that people on J1 visa experience little or no difficulty moving on to fellowship training after residency, some folks on H1 can find the going a bit tougher.
Most big, university programs have no issues when it comes to sponsoring visas. Programs at large universities are looking for excellent candidates and they don’t care about visa issues in general – they have the resources, both human and financial. If they find a great candidate on J1, they’d take him. Ditto for H1B.

