I want to be able to have a field display all of the attachments on a JIRA issue in a pick list where someone could select one and provide it to an external system.
I believe you will have to code your own to do that kind of feature for JIRA. I'll start by looking at how to develop add-ons on JIRA doc (https://developer.atlassian.com/jiradev/jira-platform/building-jira-add-ons).
Alternatively, have a browse at the Marketplace (https://marketplace.atlassian.com/search?q=share+attachment)
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