TL;DR
Add an extension building for the Rocket Silo similar to the Cargo Bay building for space platforms.What?
The PAB (Payload Assembly Building) would be an attachment for the rocket silo similar to the Cargo Bay for space platforms. It must be placed next to a Rocket Silo or another Payload Assembly Building and merges with it. The building replaces the cargo aspects of the Rocket Silo in all forms other than starting a new platform or traveling to space. Those two things remain in the Rocket Silo.The PAB has a selector for which platform it services. The default is any unassigned platform, meaning one without an PAB configured for it. This mirrors what the rocket silo currently does. This comes with some restrictions and problems just like with the Rocket Silo. For example reading requests doesn't tell you where the request is from. No way around that. This changes when the PAB is configured for a specific platform. Then the requests are only requests by that platform and cargo only travels to that platform. Want to handle 2 platforms: build 2 PABs.
Note: A priority setting like with train stations could be useful with multiple PABs. When multiple PABs have a payload ready to send the priority decides which goes into the next rocket. Without that round-robin of the PABs seems like good enough.
The PAB has an inventory of size 40 or maybe 60 with no weight restrictions and basically the same features as a requester chest for requesting items and trashing excess and such. Additionally it has the "Automatic requests from space platform" toggle for sure and maybe a "Request full rockets only" toggle that will only request items if a rocket can be filled by the current logistic network contents. Without the later the PAB will request everything configured and slowly collect a payload. The PAB also has a 'Launch' button or can be set to launch a) when full, b) when construction requests are satisfied, c) when all requests are satisfied or d) on circuit condition. The 'Launch' button sends the rocket to the configured platform or asks which platform when unconfigured and more than one platform is in orbit.
Now comes the /assembly/ part of the PAB. When a rocket is waiting on the Rocket Silo the PAB checks the intersection of requested items and available items sorting them and if a rocket can be filled fully or with too little space for any of the unavailable items then the rocket is filled and launched. Sorting items takes construction items first, turrets and ammo first, then scaffolding and then the rest by weight. Non construction materials are sorted first by anything making up a full rocket and then by weight. This means a blueprinted platform will get it's defenses up quickest so it isn't destroyed during assembly. The chosen payload might not use space optimally but it avoids the packing problem and tries to fill holes with smaller items.
Note: non construction materials could avoid breaking up items that make up a full rocket where possible. So if there are 1000 copper plates available (and requested) and the payload has 1 furnace for construction already in it then the PAB would rather fill in 30 assemblers and 19 inserters than split the 1000 copper plates. But that's purely a preference and optimization of the packing.
The PAB should have a box for the current payload plan. The box should have 2 parts: First an empty row (expands down when full) where the player can manually place items. Below that automatic slots where it shows the current content sorted by the above and cut off by the weight limitations of the rocket. Below this the "X kg of 1 ton" text can appear. When you manually hit the Launch button then whatever is shown in the box will get put into the rocket.
