I managed to resolve the problem.
The returned error "The server rejected the request because the client has exceeded request limits."
refers to "Character limits per hour" - which seem to be enforced on a sliding window per minute. See the linked documentation page
With my "F0 Free" subscription tier, the hourly limit is 2 million characters, i.e. around 33,000 characters per minute. As my single request has more than 33,000 characters, it is rejected right away.
I've tested this with sending three (identical) requests of around 13,000 characters each right after each other. The first two get translated, the third request - which brings the total above 33,000 - gets rejected with the above error message.
After waiting for a minute I get the same behavior again.
The documentation on the linked page is somewhat vague on the "Character limits per hour":
The hourly quota should be consumed evenly throughout the hour. [...] You're likely to receive an out-of-quota response under the following circumstances:
You've reached or surpass the quota limit.
You've sent a large portion of the quota in too short a period of time.
The error message could also be improved, pointing to the "limits per hour", especially as these limits can supersede the per request limits documented above.