A large discussion, but a short answer - teleportation to the build storage is not available. There is workarounds, but first I'll give a more detailed answer as to why not.

This is not a technical implementation problem, but a limitation by design.
1) The existence of dispatcher/catcher modules is the customization you mentioned, and no more customizations are needed.
Actually, the presence of a large number of conditions in the code is the result of debugging the logic and making the most transparent version of it (for a player, not for a modder).
The reverse path is full control of the ware streams, that is, in the UI you need to make a copy of the logical view for stations, but for teleportation.
[ As a developer, I really don't like messing with the UI, so I don't want to go this path, especially considering that you need to write 3 times more code than there is now. ]
2) From the game's point of view, the build storage is a separate mini-station standing nearby. Its design is predetermined and does not change.
If it were possible to add a catcher module there, everything would work now and would not conflict with the logic of the game world.
3) Game performance. Checking all the constructions all the time is too expensive.

Now, the workarounds. I use two:
1) A separate small trading station with a catcher module in the same or adjacent sector as the target station. Then use a supply mule in dedicated builder mode.
2) If the target station does not produce construction materials (restriction - 'inbound' teleportation of pure products is not available), then teleport the materials to the station and then use mass traffic to move them to the build storage.
^ Let me clarify. Mass traffic is a cut-down, but still working atavism from X Rebirth.
Moving goods using cargo drones. There must be at least one cargo drone at the station.
Then, from time to time, the station will move goods from its main storage to the build storage. There is buy (for teleportation) and sell (for moving) offers should be present at the station.

引用自 arshiba：

This is not a technical implementation problem, but a limitation by design.
1) The existence of dispatcher/catcher modules is the customization you mentioned, and no more customizations are needed.
Actually, the presence of a large number of conditions in the code is the result of debugging the logic and making the most transparent version of it (for a player, not for a modder).
The reverse path is full control of the ware streams, that is, in the UI you need to make a copy of the logical view for stations, but for teleportation.
[ As a developer, I really don't like messing with the UI, so I don't want to go this path, especially considering that you need to write 3 times more code than there is now. ]
2) From the game's point of view, the build storage is a separate mini-station standing nearby. Its design is predetermined and does not change.
If it were possible to add a catcher module there, everything would work now and would not conflict with the logic of the game world.
3) Game performance. Checking all the constructions all the time is too expensive.

What's interesting is the fact that I'm seeing random teleportation work. And this doesn't just apply to the building vault, but to everything.
Since the game developers went so far as to make the construction vault a miniseries that can hold millions of cargo, maybe you could make an exception for that too? Add teleportation to construction vaults please. That's the only reason I put your mod in, hoping it would work. But alas, I was greatly saddened to find that it didn't. However, the station that produces batteries can teleport to the construction depot itself to transfer batteries to the construction depot. But again - it's random. Sometimes it does, sometimes it doesn't.

I hope that someone will make a real teleportation of goods to construction depots in the future. Even if it will be expensive in terms of resources.

引用自 arshiba：

Now, the workarounds. I use two:
1) A separate small trading station with a catcher module in the same or adjacent sector as the target station. Then use a supply mule in dedicated builder mode.
2) If the target station does not produce construction materials (restriction - teleportation of pure products is not available), then teleport the materials to the station and then use mass traffic to move them to the build storage.
^ Let me clarify. Mass traffic is a cut-down, but still working atavism from X Rebirth.
Moving goods using cargo drones. There must be at least one cargo drone at the station.
Then, from time to time, the station will move goods from its main storage to the build storage. There is buy (for teleportation) and sell (for moving) offers should be present at the station.

Then “teleportation” itself loses its meaning as an item of use, if you have to get so worked up about it.

In X3 Terran Conflict there were cargo distribution networks from Logain Alber
https://forum.egosoft.com/viewtopic.php?t=227813&postdays=0&postorder=asc&start=0

Even there it was needed. And in X4 even more so)