Yes, the desired behavior is "writeback caching".
However, I don't think either of these systems provide a configurable threshold like "max 5 minutes of dirty cache". Instead, you might have to guess as to how large of a cache device you can risk. The larger the cache, the more volatile data might be lost. But, we generally don't have guaranteed writeback throughput either. So you cannot be sure how long a given cache size will take to fully drain to the second storage layer in the worst case.
When I have used the LVM cache mechanism, I made a RAID1 mirror of two SSDs for the writeback cache layer, and hoped they were different enough devices so that they wouldn't simultaneously fail from the mirrored traffic patterns. In fact, I used a RAID1 of SSDs to writeback cache a RAID1 of HDDs.
But these days, I usually just try to have a large enough SSD to hold my entire data and then use a scheduled backup tool like restic. It copies data to another device but also maintains an archive of multiple snapshots for point-in-time recovery scenarios. I accept up to 24 hours of loss here, using a daily backup procedure.
For really important data where I worry about 24 hours, I might manually replicate after making important file updates. E.g. git push to replicate commits to a remote repo, or rsync to replicate specific subdirs to another machine.
Edit to add: when worrying about 24 hours, it is also worthwhile to think about the probabilities. The extra effort to maintain a low latency backup might end up costing more time than the unlikely potential day of redundant work after a loss. And how long will the recovery take anyway? Do you keep spare parts on hand or expect to spend time procuring hardware and rebuilding your system after a period of inaction due to a broken system...?
The things that I consider too important to wait 24 hours would be non-deterministic things where a recovery method is more difficult than just performing the day's tasks again. E.g. losing recently changed passwords/secrets in an encrypted password store, or losing my notes about other infrequent purchases or bill payments that are in progress. I wouldn't just repeat the steps, but instead have to unravel other recovery procedures for each item.