This is a real issue with Meraki, not just the OP, we are experiencing the exact same issue with only clients connected to a Meraki Client VPN.
This seems to be Microsoft's fault at the core, not having a switch to register DNS when the connection is created on the client. I haven't seen a real solution, just workarounds.
Going to test adding this script to my Add-VpnConnection script.
$RASPhoneBook = "C:\ProgramData\Microsoft\Network\Connections\Pbk\rasphone.pbk"
(Get-Content $RASPhoneBook) -Replace 'IpDnsFlags=0', 'IpDnsFlags=3' | Set-Content $RASPhoneBook