VxLAN offers a lot of flexibility, however it might not be the best answer in this case. You may be better off using VPN tunnels between both cloud environments. That is, assuming you can have multiple VMs within the tenant on those providers, have them point their default gateways towards a VM within your control, and use that VM as a firewall/VPN concentrator. From there you can establish a S2S VPN between the cloud environments, and can NAT the traffic from your provider's WAN IP address to the appropriate host, whether locally, or at the remote environment.
If you must have L2 connectivity between your cloud environments, I can speak from experience only in a Juniper environment, and in that case we would place a vMX VM behind a vSRX VM. The vMX VM would act as the EVPN/VXLAN VTEP, and your VMs would set this as their default gateway. The vSRX would establish IPSEC S2S tunnels, through which data-center interconnect (DCI) traffic would flow. L2 traffic would flow through the vMX, where it would be encapsulated in a vxlan tunnel, which would route through the SRX, which would then encapsulate this in an encrypted IPSEC tunnel, before sending to the other data center. Details of this might be a little too complex for a stack exchange answer though.
Hope this helps point you in the right direction!