In your save button click event, call the built-in jira rest API with AJS.$.ajax() and supply the value for your custom field.
var issueKey = JIRA.Issue.getIssueKey()
var valueFromForm = '' //get the value from your form, perhaps using AJS.$('selector')
I think you found the relevant post that already covered this: https://community.atlassian.com/t5/Adaptavist-questions/Display-Custom-Web-Items-in-a-Dialog-from-Issue-Actions-on/qaq-p/1635553?utm_source=atlcomm&utm_medium=email&utm_campaign=mentions_reply&utm_content=topic#U1747825
Hello Community! Quick disclaimer: We are running a contest on Community (The Atlympics!) from July 23rd - August 8th of 2021. If you are interested in participating in this contest (prizes! ...
Connect with like-minded Atlassian users at free events near you!Find an event
Connect with like-minded Atlassian users at free events near you!
Unfortunately there are no Community Events near you at the moment.Host an event
You're one step closer to meeting fellow Atlassian users at your local event. Learn more about Community Events