I emailed them regarding whether any TLC trusts cannot offer tier 2 visas, and got this reply:
Dear ______
Thank you for your email.
All trusts are in the same position and there are no trusts which cannot offer Tier 2 visas.
Please note, an employing Trust will only be eligible to provide a Certificate of Sponsorship to allow you to apply for a Tier 2 visa if you are either currently completing an Mpharm or OSPAP MSc. Please be aware that they will not be able to do so if you are currently studying an OSPAP PgDip.
I hope this is helpful.
Best wishes,