I do not believe it can be done.
I have encounterd the same problem. SSRS will recognize fore color. It will not recognize background-color or even the tag level bgcolor
i ran a quick test to see if there was anything i can do in ssrs to get around the limitations. It appears that SSRS actively scrapes the content and the styling and then applies it as it wants.
Here you see a quick report I made. I entered some simple HTML into a table and then tried to display that as HTML. You can see the raw HTML and the resulting SSRS rendering. The first column is Len(rawHTML). There was a question as to whether SSRS or SQL was truncating the rawHTML. The column is currently defined as nVarchar(8000).
Here is a peek under the covers. This is CHROME DevTool showing the rendered HTML for the NOT UNDERLINED tag from above. You can see that none of the actual HTML is passed through.
MS seems to have very strong feelings about not allowing HTML injection or JS injection into reports. ( I have tried everything I can think of to get a report to import a D3 image from a webpage, I have not been able to do it)
I can begrudgingly give to MS that allowing JS injection would lead to an avalanche of error reports. To a lesser degree, i can see HTML insertion being a potentional problem.
Unfortunately, this leads to their 'scrape and emulate' approach. The support for HTML formatting seems to be pretty shallow.
For those interested, here is the version information for my SSRS. SQL Server is 2019 Enterprise.