Ok, so I found a solution. To clarify, what I want to do is wake up the NAS with a 1gbit connection, that is then not used for anything else anymore.
Changing the order of cards or default gateway status in DSM by itself did not help. Windows still routed all traffic going to \\NAS_Name\ through the slow IP/network port.
The trick to get it to work is to remove the slow connection from the network completely, as all the port needs to receive is the magic packet for waking up, which is connected with the MAC address, but not the IP address.
To do this, after one has set up Wake-On-LAN in Synology Assistant, one just has to completely screw up the network settings of the LAN port used for WOL. I disabled IPv6 completely, and for IPv4 I disabled DHCP, and set the IP address to 10.1.2.3, the subnet to something different from my normal subnet, and deleted the entries for Gateway and DNS server.
Now I can wake the NAS up, but Windows only knows the 10gbe adapter (that still has DHCP), but not the slow adapter with the messed up settings. So all traffic is nice and super fast as it should be.