As you discovered, the <<Report message does not work on a Dashboard. JMP Applications in general can have multiple windows, though a Dashboard will generally only have one. The <<Get Windows message is similar to <<Report, but it will return a list. This would be an alternative to using Current Report(), with the assumption that there is only one window:
Open("$SAMPLE_DATA/Quality Control/Steam Turbine Historical.jmp");
app = JMP App();
app << Open File( "$SAMPLE_APPS/Instant App.jmpappsource" );
box = app << Run;
(app << Get Windows()) << Save interactive HTML("$TEMP/app.html");
Open("$TEMP/app.html");