Using the HTML5 translate attribute might work, but yes, it would assuredly be very labour-intensive.
The general quality of external translation services seems to vary. I’d suggest the onus is really on those services to avoid inadvertantly translating CSS code along with other content.
At least, that seems easier than the alternative of educating content creators to stop using inline CSS within the HTML.
Separation of concerns in coding web content has been greatly encouraged since the early 2000’s, yet a significant portion of content creators (individuals and publishing systems) still remain ignorant or simply ignore it.