It’s not all that easy to make any clinical diagnosis of precisely where a particular noise is coming from when - inherently - the thin, sheet steel of which Lambrettas are constructed act as a drum.
I can only suggest that with panels off, any rattling items - including toolbox lids, contents & panel clips secured or bungees attached, then the engine be fired up & the tickover adjusted to fast. The most likely unsafe sounds will be knocking or ‘pinking’ but there are likely to be sounds that you can identify to eliminate from your list of major concerns. Unfortunately, the OEM exhaust mounting is rigid, which method in itself contributes to noise. It can be improved & virtually eliminated via mounting modifications, but that may be for another day. The air filter box is also a source for noise. Even the rear shock absorber - unloaded as when the bike is on it’s stand - will often contribute. The stand & splash plate, tight chain, crankcase cover/kickstart lever catching the clutch, etc etc. A tool rarely mentioned these days is the ‘one eared stethoscope’ which is simply a long piece of fuel pipe tubing by which to listen in by holding to various parts of the engine.
I will venture to say that it is often the case that a replacement product from a trusted specialist supplier will not be on the same page as the OEM item. Replacement part quality not only affects us, but there are a lot of companies producing identical looking components in the car industry which will fail, as I can attest, having fitted a pair of rear upper shock absorber mounts to my car in January that made it as far as June. Having worked for some periods in the industry in which engineered, ‘rubber’ based automotive components are produced, I have some knowledge of the processes involved. Besides the ‘rubber’ being of a base consistency to endow desired properties to the finished component, the bonding of the materials involved will be so good that the component will likely fail for another reason prior to the integrity of the bond. In ‘proper’ anti-vibration components, such as a Lambretta engine mount, there will be additional processes after moulding to deliberately add desired properties to the ‘rubber’ by stressing it. In the case of the OEM engine mounts, that will have almost certainly have been to ‘swage’ the steel tube bore by pressing a tool through to give it the ‘finished’ size. Likewise, the spigot - in the case of the larger engine mounts - will be reduced from ‘moulded’ size to it’s finished size.
As far as I am aware, when Innocenti & it’s trusted suppliers ceased producing OEM components, probably the licensed Spanish suppliers & certain of the SIL parts manufacturers continued with the same methods of producing engine mounts. I think it would be naive to believe that without some kind of proof, or basis for trust, that reproduced engine mounts are going to be of the same quality. Imagine the costs involved in just the tooling for hydraulically reduced the spigot size

Unless things have changed, the last of the ‘two hole, no void’ engine mounts - despite being ‘solid’ rubber - are highly prized by racers & - probably! - more ex-racers, but that will be for reasons of handling improvements.
Sorry to have waffled on, but I’m attempting to paint a fuller picture as to the scenario as I see it. Don’t lose heart about this ‘teething’ problem. It’s far too easy to blame the scooter itself, but, of course, the real culprits inevitably will be those that set up in business selling us stuff that we trust in. That’s not to say that they are all alike, but they all exist to put groceries on their table….
