Your issue is definitely that the switch doesn't support LACP. You can try enabling Static Link Aggregation or just Fault Tolerant links at the NIC level. The latter won't give you any performance benefit, though. And, as has been mentioned, you still have the option of using the built-in NIC teaming support in Server 2012 R2.
However, one thing I would do is enable Hyper-V and install a second OS on the server for file sharing. You can assign that VM it's own NIC so your workloads will be spread out. On Server 2012 R2 Standard you can run up to two VMs without additional licenses. Server 2012 R2 will do a check during the install to see if the host OS is Activated and if so won't prompt for activating the new VM.
If they are Intel NICs, be sure to enable SR-IOV when you setup the virtual switch(es), and then enable that for the VM. It'll give the VM direct access to the NIC instead of having to go through the hypervisor. For just file services you can get away with assigning just 2 vCPUs and 4GB of RAM. If you think you need more RAM (which you shouldn't for file services) you can assign a dynamic amount with the minimum being 4GB and determining an upper limit of something like 8GB, then let Windows/Hyper-V assign as required. I've got DHCP servers that drop down to just a couple hundred MB with dynamic memory assignment.
Another option may be adding an additional IP on one of the NICs and configuring the database to talk on that IP address.
There are a multitude of ways you can configure the link(s). Honestly though, if speed is that paramount you need to look at getting a good, managed, non-blocking switch for the server to connect to.
As for maxing out a GigE link, a GigE Windows 7 or Windows 8 client going to a GigE Server 2008 R2/2012/2012 R2 server can most definitely max out a single GigE link if transferring large files.
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