Hello,
From your diagram, it seems like you have your network/sites setup to get DHCP from the mitel controller already.
Assuming you are using routers to reach your remote location - and the links b/n sites are in the same subnet...it should just work. As long as you setup your ACLs, static routes or routing protocol(s) correctly.
However, you can configure DHCP on your remote site router for the 'phone' network.
Regardless, I would strongly recommend you deploy QoS over the link - so the phones don't saturate it. Although you only have 10 phones, and assuming dial tone all comes from the CO - voicemail, and other services can saturate the link under certain circumstances.
Regards,
Mike G.