TeaRex, if it's on a trust, what does the CLEO have to do with it? Especially in Texas? A 7-day wait? Just curious to see what kind of BS has happened in my home state while I'm stationed elsewhere...
To the OP, if SS said they'd file your Form-4 for you, then that's between you and them. My understanding is that the dealer they Form-3'd it to will fill out and file the Form-4 after they have a suppressor in hand, and not sooner. Perhaps SS just collected your trust info so they can pre-fillout the Form-4 and send it with the suppressor so your dealer has little-to-no excuse to not file it right away? If I were you, I'd clarify with SS. They are great folks and should walk you through the details.
I purchased a suppressor from them, and they are fast. Form-3'd to my dealer as soon as possible. Shipped it to him fast. We picked up the Form-4 process on our side.
As for the waiting for the Form-3 before shipping, yes. The Form-3 is the BATFE's way of saying that the ownership has transferred from SS to your dealer. If SS sent the suppressor before they were sure that the transfer was 100%, then there's the risk that the Form-3 gets rejected for some reason, and now the dealer would be in illegal possession of a suppressor they don't own, and SS would have recklessly lost possession of a suppressor in their inventory. Likewise, the dealer should wait until the suppressor is in-hand before filing for the Form-4. If they do it early, while the Form-3 is still processing, what if the Form-3 is rejected? Then the dealer would be filing as "Transferor" to transfer a suppressor they don't own, to you, the "Transferee". In the grand scheme of things, just wait the extra few days to make sure everything is correct before filing the next step.
For what it's worth, double/triple-check that your trust name is spelled correctly on the Form-4. Mine was wrong on one suppressor from a different source than SS (e-Forms submission, so I never saw it), and it still got approved from the examiner. I noticed the mistake when my dealer forwarded me my Form-4 pdf. I didn't want there to be any question that my trust was the legal owner and not some other trust, so I had my dealer point out the mistake to the BATFE rep he deals with. They took about a week to get it all squared-away, with a new-and-correct Form-4. I'm fine with the extra time to get it right. Peace-of-mind, and all that...
Good luck with your purchase. Welcome to the addiction!