I got some feedback from the Synapse Product team and it turns out this is a known issue and there is yet to be an ETA to have it fixed.
See below an extract from the feedback:
Currently the synapse studio doesn’t show the connected cosmos DB if the connection string is from Azure Key Vault. We have plans to introduce a message/warning to users stating ‘ Cosmos DB won’t be seen under ‘Data’ section if Azure Key vault is chosen.
So for anyone who is wondering, currently if you want the CosmosDB link to show under 'Data' do not use a Key Vault backed Linked Service.