Hi Martin,
It's a little while since I've worked with SALSA, but I did have a quick look at the new Issue 6 standard and associated guidance notes. Unsurprisingly it seems to have moved forward a little towards the direction that BRC has gone, albeit still very much "lighter" than full BRC.
From my reading of 1.6.2 and the associated guidance notes, I can't readily see how you could accomplish this - there is clearly an expectation that specs are seen and reviewed prior to purchasing, and that isn't going to be possible buying from a supermarket. If you make your own spec there is a legitimate question as to whether it has any real value - the supplier isn't bound to it, and doesn't have a mechanism to tell you if their product changes with respect to the spec.
You could try asking SALSA themselves, as having worked in that scale of facility, I'm very sure you're not alone in purchasing from this type of source. I'd understand if SALSA weren't keen on it though, as it might be difficult to demonstrate suitable control - if e.g. the allergen status of one of the products changes, how do you check this?
You may want to explore buying from some of the foodservice or cash and carry suppliers instead - the quantities will still be smaller than would be the case with full industry suppliers, but many of these now have systems to provide you with an actual specification for products etc.
For some types of materials there are also online suppliers who can and do provide specs and CofA for very modest quantities (e.g. additives sold in quantities in some case <1kg) that would allow you to fulfil the requirements more easily, whilst also giving you slightly more control over / knowledge of the nature of the raw materials.