Yes I've just tried to do this in Linux, the "route del" or "ip route del" command is accepted without an error, but the AnyConnect default route doesn't go anywhere.
The only workaround I can think of is virtualize the IP stack that the AnyConnect client is running under = either full-fledged HW virtualization or at least something like a docker container... which obviously has other practical downsides, as you are effectively encapsulating the whole machine where AnyConnect is running.
At a political level, perhaps it's better to stick to the "agreement" with the AnyConnect system admin that you are entering by using the VPN session. If the admin says no, the right thing to do is just refrain from workarounds...