Formula URL or Formula Text Button that will perform an action, without the need to navigate away from the current dashboard/report/record?

To test this, I suggest that you make a temporary formula URL field with just this.

URLRoot() & "db/" & Dbid() & "?act=API_EditRecord&rid=" & ToText([Record ID#]) & "&_fid_10=1"

My guess is that when you click that it will fail on something, likely a complaint about the application token is missing.

I suggest that you then go to the settings for the App  and then Advanced settings and either disable the need for Application Tokens or else create one and include the extra line

URLRoot() & "db/" & Dbid() & "?act=API_EditRecord&rid=" & ToText([Record ID#]) & "&_fid_10=1"
& "&apptoken=xxxxxxxxxxxx