To do it you have to turn the P2P circuit into an internet circuit.
So lets take a simple case. Lets say at the remote site their is an ISP supplied modem with four ports doing NAT on the LAN side (so you can just plug in a computer and access the Internet). You have the remote MX plugged into one of those ports. Move the P2P circuit so that it also plugs into this ISP supplied router.
Now on your site move the P2P circuit to WAN2 on the local MX. Enabled AutoVPN. Set the preference so that VPN traffic prefers WAN2, and Internet can fail over to it.
So traffic between the two sites will flow over AutoVPN over the P2P circuit between the MX WAN ports. The local site can now also fail over its Internet to that of the remote site.