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See, I've GOT SL3, but why use that if I've got the Rane 56s?
because the SL3 has the exact same phono pre-amps as the 56 (might be even better) so you can exclude the mixer from the chain and it'll still sound the same. In fact, there's probably less degradation to the signal because that's one less volume/gain chain to go through than if you record directly into your SL3.
What you're asking for right now is going turntable -> 56 -> recorder -> CD and then export those files to the laptop.
With the SL3, you can go Turntable -> SL3 -> Laptop and then if you want to have a hard copy, then burn your archived files to CD. It's actually a more direct signal path to an audio recording and you don't lose the analog sound because the SL3 sports a pair of analog phono preamps (this is not the case with the SL1 however).
Your issue with gains - even with a recorder and the 56, you still need to deal with levels and gain structure - first on the 56, and then on the recorder.
With the SL3 path, you have one gain to adjust within the Scratchlive software and that's it. You don't need a 3rd party piece of software, and USB latency isn't an issue since you're just recording for archiving purposes - latency only matters for manipulating the audio in realtime.
Personally, I also wouldn't worry too much about laptop/computer as a problematic platform considering that virtually every studio these days have gone to computers for all their recording save for a few novelty acts that still demand analog.
If fidelity a high concern, you can always record in 24-bit, but keep in mind, you'd have to convert your files to 16-bit audio in order for serato to play (this would still be the case if your digital recorder supported 24-bit).
To be honest, I think things like needle choice, turntable noise, and the shape of your vinyl is going to have more of an impact on the sound quality of your rips rather than whether you choose to record onto a computer or if you use a hardware recorder...
