If I recall correctly ( my experience is getting old, it's from 2019 ) I showed outbound ticket. One thing the ticket was needed for was to determine number of days in storage, I believe there was at least two tariffs, one for short term and one for long term storage. Some items are not allowed to be stored more than x days and they will need to know departure day in advance for these two reasons at least.

In my case, I had a big bag meant to be checked in, but I don't know how much of that was handled when storing the bag. I received some kind of ticket and when I departed TPE I showed it to the check-in-desk and they arranged bag-tag and delivery to cargo hull. Delivering that ticket to check-in was crucial for that to happen, ie your actual departure is what triggers bags to be removed from storage and checked-in. For a check-in bag, the airport handles bag from storage to cargo-hull, I never touched the bag until reaching final destination.

Based on this (and if memory serving me right and things still work the same) I guess your method could work technically as long as you show the second outbound ticket and have bag contents that is allowed to stay in storage for that number of days.

In reality, I am not sure they would appreciate keeping something in store while you leave the country.
Also, the storage is allowed to hold items that are prohibited in Taiwan and I think you are exposing yourself to a number of issues experimenting with the limits of the service. In your case, I would just bring the bags into Taiwan and use a storage service landside. Slightly less convenient but much better for everyone.