Router Managment Access when interface is down

Question

Hi,
Please see the topology attached.
We have a customer network with number of routers/switches. We have a management network to manage devices via telnet/ssh.
On switches we have a vlan interface for switch management  while on routers we have sub-interfaces ( 802.1Q trunk, with encapsulation) connected back to the switch for the management.
Problem :
Customer has ask us to give them a access to router/switches, we have give them telnet/ssh access via management network, to access router remotely customer ssh router (the sub-interface IP address F0/0.10 on router), but when the router interface Fa0/0.10 is down ( because switch at the remote end is down), customer cannot  the telnet/ssh to router.
How can I allow customer to keep accessing the router while sub-interface on the router is down ( which they are telneting to)? I am happy to change to router config, but not sure which bits.
I can't create the loopback interface and assign the IP address to it from the managment network as the router subinterface F0/0.10 is already have IP address from that subnet and router gives overlapping mask error message.
I created the new looback interface on router and give is the same IP as of F0/0.10 and configure  F0/0.10 as a IP unumbered loopback 0,  it;s not working either for me.
Can I somehow configure the router to respond to the telnet/ssh when subinteface is down- I am happy to move the addresses, create new interfaces , change routing etc. but I can't change the network subnet that is already assigned to customer.
Please see the topology attached.
Any idea from anyone.
Regards

Answer 1

Thanks for your responses.
I don't want to allocate the new subnet with /32 for the management as it will require many changes in the network such firewall  etc.
There will be a single  switch connected to the router physical interface F0/0, but there will be a multiple switches hanging off the first switch. ( all switches in the vlan10, including router sub-interface F0/0.10).
Customer will require access to both, switch(es) and router, customer understand that if the first switch ( that physically connects to the router interface F0/0 ) fails, access to all other switches will also fail, which is acceptable. At this point we must have access to router regardless we have lost access to the switch.
Customer want router to be accessible even if the switch(es) are down, as the router at the point router is fine and is still connected to the WAN network. Customer will lose the access to the switch(es) but should not lose the router access.
We have different IP subnets ( VRF's) for the customer data network ( LAN) and the router management, so I can't assign the router management IP address from the customer LAN subnet
Forgot to mentioned that we have three VRF's on router ( vrf-lite/ multi vrf) , one for customer data network, one for router management, one switch ( es) management.
Fa0/0.10 is in the switch management VRF, while  router Loopback 0 is in the router VRF.
We have to maintain the vrf's to keep router and switch management traffic separate.
Router is always accessible  to us ( not to customer) via router vrf hence its still available even if the router LAN management interface F0/0 is down. 
Customer lose the access to both router and switch(es) if the F0/0  down.
The only option I can see would be to allocate a new subnet for customer router management and assign this to a new loopback and put under the switch management vrf.
Regards