I think it's good that you're considering posting the update directly to UKVI, that's what I would personally do as well to avoid any further delays. However, make sure you explicitly mention in your covering letter that your application was done through UKVCAS and that they told you get in touch with UKVI directly, as per the email you received.
The reason for this is that I personally cannot confirm that the address UKVCAS offices use to send out applications to UKVI is the same as for other applications. The address in Liverpool is likely to be the same, but there are different departments that receive the applications. So they'll need to know where to forward it if it arrives at the incorrect department.
For example, paper applications are posted to Department 1, as observed by yourself. I personally had to post my supporting documents (online, non-UKVCAS) to Department 4, and I received my approval letter along with the supporting documents from Department 71 (mentioned in the return address).
On a side note, I find the UKVCAS reply you got rather disappointing given that they are the intermediary and it is their responsibility to be handling your application. If I were in your shoes, after posting the update to the application to UKVI I would also put more pressure on UKVCAS, or at least file a comment/complaint on your experience with the new service and how it can be improved. IMO it's completely inappropriate for them to be telling you essentially "you sort it out with UKVI we've done our part". If you have to deal directly with UKVI for certain aspects (e.g. updates to application) I feel it is UKVCAS' responsibility to at least tell you exactly how to do that and where to post any correspondence if necessary.