In Power Query I wrote a simple function that calls a web API to get the local time for a country.
I then use function in a table of countries to get all the times for different countries. I call the function while adding a column with one of the fields in the column as a parameter - CountryCall is the field and CountryTimeFnx is the function.
I had a look at Chris Webb’s blog from the link you’ve sent and tried it. However in the blog, query parameters are used. The initial API I called “World Time API” (http://worldtimeapi.org/) does not use query parameters and so this strategy doesn’t work.
I found a different API “Time API” (https://www.timeapi.io/) that does use query parameters and tried that out with the techniques used in Chris’ blog. That did the trick and it now allows refreshes when published to Power BI Service.
For reference I’ll link the updated pbix here, if anyone needs that as a solution to similar issues.