Using the Rest API to extract the virtual server's advanced settings

How much info do you need....? You can add the expandSubcollections query parameter to your call to expand the config returned - https://x.x.x.x/mgmt/tm/ltm/virtual/vip_name/?expandSubcollections=true

Or if you want output like tmsh list ltm virtual vip_name all-properties you can run the command using the api bash util as detailed here - https://devcentral.f5.com/s/question/0D51T00006i7fLs/native-tmshbash-commands-via-rest-api

So call would be something like

curl -sk -u 'admin:admin' -X POST -H "Content-type: application/json" -d "{\"command\":\"run\", \"utilCmdArgs\": \"-c 'tmsh list ltm virtual vip_name all-properties'\"}" https://x.x.x.x/mgmt/tm/util/bash | sed 's/\\n/\n/g'