Well, a bit of progress has been made.
This flow works.
[{"id":"fea1dbc8.8b4ad8","type":"function","z":"cc5873bb.2070e","name":"","func":"msg = {payload: '<font color = \"yellow\" i class=\"wi wi-sunrise fa-2x\"></i>'};\nreturn msg;","outputs":1,"noerr":0,"x":520,"y":640,"wires":[["a8c18cd3.7209a8"]]},{"id":"98e47d10.0fb678","type":"inject","z":"cc5873bb.2070e","name":"Sunrise","topic":"","payload":"sunrise","payloadType":"str","repeat":"","crontab":"","once":false,"onceDelay":"","x":390,"y":640,"wires":[["fea1dbc8.8b4ad8"]]},{"id":"a8c18cd3.7209a8","type":"ui_text","z":"cc5873bb.2070e","group":"3bc589f2.8828ce","order":3,"width":"2","height":"1","name":"","label":"","format":"{{msg.payload}}","layout":"row-spread","x":660,"y":640,"wires":[]},{"id":"e70d7395.25d7c","type":"inject","z":"cc5873bb.2070e","name":"Night fog","topic":"","payload":"night-fog","payloadType":"str","repeat":"","crontab":"","once":false,"onceDelay":"","x":380,"y":680,"wires":[["17b11034.bf65a8"]]},{"id":"c87f4586.abf148","type":"inject","z":"cc5873bb.2070e","name":"Day sunny","topic":"","payload":"day-sunny","payloadType":"str","repeat":"","crontab":"","once":false,"onceDelay":"","x":380,"y":720,"wires":[["27335178.4cafe6"]]},{"id":"8e3c20f1.67632","type":"inject","z":"cc5873bb.2070e","name":"Day fog","topic":"","payload":"day-fog","payloadType":"str","repeat":"","crontab":"","once":false,"onceDelay":"","x":390,"y":760,"wires":[["49728963.cc3778"]]},{"id":"20b63b62.46160c","type":"inject","z":"cc5873bb.2070e","name":"Day rainy","topic":"","payload":"day-rainy","payloadType":"str","repeat":"","crontab":"","once":false,"onceDelay":"","x":380,"y":800,"wires":[["f66462ea.3c62a8"]]},{"id":"17b11034.bf65a8","type":"function","z":"cc5873bb.2070e","name":"","func":"msg = {payload: '<font color = \"yellow\" i class=\"wi wi-night-fog fa-2x\"></i>'};\nreturn msg;","outputs":1,"noerr":0,"x":520,"y":680,"wires":[["a8c18cd3.7209a8"]]},{"id":"27335178.4cafe6","type":"function","z":"cc5873bb.2070e","name":"","func":"msg = {payload: '<font color = \"yellow\" i class=\"wi wi-day-sunny fa-2x\"></i>'};\nreturn msg;","outputs":1,"noerr":0,"x":520,"y":720,"wires":[["a8c18cd3.7209a8"]]},{"id":"49728963.cc3778","type":"function","z":"cc5873bb.2070e","name":"","func":"msg = {payload: '<font color = \"yellow\" i class=\"wi wi-day-fog fa-2x\"></i>'};\nreturn msg;","outputs":1,"noerr":0,"x":520,"y":760,"wires":[["a8c18cd3.7209a8"]]},{"id":"f66462ea.3c62a8","type":"function","z":"cc5873bb.2070e","name":"","func":"msg = {payload: '<font color = \"yellow\" i class=\"wi wi-day-rainy fa-2x\"></i>'};\nreturn msg;","outputs":1,"noerr":0,"x":520,"y":800,"wires":[["a8c18cd3.7209a8"]]},{"id":"81def411.cf43","type":"ui_template","z":"cc5873bb.2070e","group":"3bc589f2.8828ce","name":"Load CSS","order":0,"width":"0","height":"0","format":"<link rel=\"stylesheet\" href=\"/weather-icons/mycss/weather-icons.min.css\">\n<style>\n .nr-dashboard-theme-dark ui-card-panel {\n \tbackground-color: #222;\n \toutline: 1px solid #cccccc;\n \t }\n</style>\n","storeOutMessages":false,"fwdInMessages":false,"templateScope":"local","x":468,"y":343,"wires":[[]]},{"id":"3bc589f2.8828ce","type":"ui_group","z":"","name":"Test","tab":"77086a97.0ed444","disp":true,"width":"6"},{"id":"77086a97.0ed444","type":"ui_tab","z":"","name":"Test","icon":"dashboard"}]
I press the buttons and the icon changes.
I cut and move part of the flow to the flow I want it on, and this is what I have:
[{"id":"e08af36a.5c84c8","type":"inject","z":"c636aa5a.cc34","name":"","topic":"","payload":"wi-sunrise","payloadType":"str","repeat":"","crontab":"","once":false,"onceDelay":0.1,"x":1540,"y":550,"wires":[["b1a29603.52fb5","a8dc54d3.6f7798"]]},{"id":"b1a29603.52fb5","type":"function","z":"c636aa5a.cc34","name":"","func":"msg = {payload: '<font color = \"yellow\" i class=\"wi wi-sunrise fa-2x\"></i>'};\nreturn msg;","outputs":1,"noerr":0,"x":1670,"y":550,"wires":[["1d62649.1375c1b"]]},{"id":"1d62649.1375c1b","type":"ui_text","z":"c636aa5a.cc34","group":"ccfdf225.95c52","order":2,"width":"2","height":"1","name":"Sunrise icon","label":"","format":"{{msg.payload}}","layout":"row-spread","x":1820,"y":550,"wires":[]},{"id":"ccfdf225.95c52","type":"ui_group","z":"","name":"Daytime mode","tab":"1c792414.600e94","order":1,"disp":false,"width":"3","collapse":false},{"id":"1c792414.600e94","type":"ui_tab","z":"","name":"Telemetry","icon":"dashboard","order":2}]
I deploy the flow and press the inject button.
Go to the web page. No icon.
Here is a comparison (all on the same machine)