not much you can do without this 3rd party assistance - And then for security reasons you would require maybe a IPSec VPN between the two sites to gain the connectivity you require.
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This might seem a little basic but could you not stick a Router between your L3 switch and their 2960?
Configure the Routers WAN port to pickup a DHCP address and plug this into the cable accessing their network.
Make sure the router does NAT so the only address they see is the one assigned to you via DHCP.
The LAN port of the router would need to be on Vlan 20 or could be on any Vlan as long as it had a static route.
Bash a static route on your L3 switch (for their IP network) to point to the LAN address of the router. Then another static on the router for the destination subnet, the next hop being the default gateway of the WAN interface.