Your existing foam sounds like a fairly standard flexible open-celled Polyurethane foam, probably with an additional black velvety fabric layer glued to the top surface.
The flex. PU foam is typically produced in large semi-continuous blocks, ca. 2m wide by 10's of meters long, and post fabricated into a very wide range of products, including mattresses, furniture, packaging, etc. It's no different in essence to the foam blocks / sheets you'll often see for sale on market stalls, for cushions etc. The grey colour probably has no great significance, it just looks nice, and masks any foam discolouration from light / UV exposure. I'd be looking for a relatively high density (say 40 - 60kg/m3), high resilience (HR), grade flexible foam, with medium to high hardness.
Cutting flex. PU foam by hand can be a challenge especially when you move away from straight lines. For smaller circular holes up to ca. 1/2" I use gasket punches, for 1/4" up to ca. 1" I use freshly sharpened cork borers. I've not had to go above ca. 1", but for a camera lens diameter I'd be looking for something like a larger cork borer type tool, e.g., these leather tools, or something in that direction - you want to be cutting the foam with minimal distortion / compression of the foam (or maximum) during the cutting.
https://www.amazon.co.uk/Leather-Cut...82901373&psc=1
The more uneven foam distortion you have during cutting, the more distortion of the final cut surface, e.g., as a somewhat extreme example - these 'egg crate' foams, cut using only a straight knife blade... acoustic pyramid type foams are made in exactly the same way.
https://www.evilmadscientist.com/2014/egg-crate-foam/
As an alternative, you could also cut square lens slots, with the width of the square equal to the lens diameter, and then 'fill' the open 4 corner spaces with glued in triangular foam sections, for a nice snug fit. It wouldn't be as snug as a cylindrical hole, but done cleanly it wouldn't be far off..
For large scale production, the fabric cover may well have been pre-formed to the necessary shape. For one-off home production... a decent spray contact adhesive, and a thinnish fabric, probably pre-cut in several pieces, seems like a viable alternative.
Overall, it'll probably come down to how complicated the internal foam geometry is in terms of cut-outs, how many, depths, etc... a built-up layered approach could be better if it's more complicated.
Alan