I get access denied error when a user with less than “Contribute” try to use “Specify your own value” . The same user can use the option in native Sharepoint forms.
I created a group and gave it custom permissions (everything below Contribute and that was Read, View Only, Approve, Manage Hierarchy,Restricted Read) and I added a user to this group. Everything worked well, I was able to use and save a custom value with no problems.