Can I use a GRE tunnel to solve my problem?

Question

Please see the attached file for a topology of the relevant portions of this network.
All but three of the APs at Building B are plugged into Cisco 3650 switches that are also acting as the WLCs.  This allows for local switching of WiFi client traffic.  The WiFi clients are tagged with VLAN 20 and the PCs at Building B are tagged with VLAN 10.  Inter-VLAN routing occurs at the 3560 in Building B.  This is important so that iPads on the WiFi network are switched locally with the PCs in the classroom. I then turn on the mDNS feature on the 3650/WLC so that we can use our PCs as "Apple TVs" via a program called Air Server.  This allows the teacher to project the iPad onto the PC, which is then projected to the SMART Board.
My problem is with the 3 classrooms whose APs plug into a 2960-PS.  These APs are managed by the dedicated WLC-5760 located at Building A.  This means that the teacher PC is using the 3560 in Building B as the default gateway while the wireless traffic is being handled by the 3750 in Building A.  The last time I checked, the WLC 5700 series controllers did not have Flex Connect as a feature.  
Here's my question:  Is there any type of IP tunneling solution I could use to tunnel a particular client or VLAN so that it can be routed at Building A?  I've only played with tunneling from an IPv4/IPv6 standpoint.  Thank you for your time!

Answer 1

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You're correct, you cannot extend L2 across L3 unless you use some kind of encapsulation technology, for example, the already mentioned L2TPv3 or pseudo-wire over MPLS, etc.
However, what I have in mind for extending a VLAN means converting a routed p2p link to a L2 trunk link (I'm assuming the equipment, e.g. L3 switches, can support this). Across the trunk, you can extend your VLAN(s).  For the routers, you can dedicate a new VLAN, across just the trunk, that takes the place of the former p2p.  I.e. so you can do both L2 and L3 across the same physical link.
[edit]
I didn't see Jon's post until after I posted above, but he's explaining, in more detail, what I had in mind.