Hi!
If your connections are A<->B and B<->C, then the simplest way to set this up is to add a point-to-point VLAN between B & C just like the one between A & B, and enable routing on site B. I would recommend moving from static to dynamic routing in conjunction with this, although if you're not expecting to grow any further then static will still work. I prefer this setup because it most closely maps the underlying topology.
Here are some (untested) config snippets for this scenario:
switch A:
ip route 10.10.0.0 255.255.0.0 192.168.1.45
switch B:
vlan 3997
untagged A2
ip address 192.168.2.45 255.255.255.0
...
ip routing
ip route 0.0.0.0 0.0.0.0 192.168.1.46
ip route 10.10.0.0 255.255.0.0 192.168.2.47
switch C:
vlan 3997
untagged A1
ip address 192.168.2.47 255.255.255.0
...
ip routing
ip route 0.0.0.0 0.0.0.0 192.168.1.46
The scenario you've described will also work; as long as A[F1], B[A1], B[A2], and C[A1] are all in the same VLAN and all have a reachable IP address on that VLAN, then they should be able to route via one another. If you find it easier to think about the two links as a bus rather than two point-to-point links, then you could go with this topology. Sample configs:
switch A:
vlan 3998
untagged F1
ip address 192.168.1.46 255.255.255.0
...
ip routing
ip route 10.10.0.0 255.255.0.0 192.168.1.47
ip route 10.40.0.0 255.255.0.0 192.168.1.45
switch B:
vlan 3998
untagged A1,A2
ip address 192.168.1.45 255.255.255.0
...
ip routing
ip route 0.0.0.0 0.0.0.0 192.168.1.46
ip route 10.10.0.0 255.255.0.0 192.168.2.47
switch C:
vlan 3998
untagged A1
ip address 192.168.1.47 255.255.255.0
...
ip routing
ip route 0.0.0.0 0.0.0.0 192.168.1.46
ip route 10.40.0.0 255.255.0.0 192.168.1.45
I think an important thing to consider here is: how will your network look if you need to add a 4th switch? Will the scenario look different depending on whether you connect switch D to A, B, or C? How will the configurations look? Given those ideas, having a bus topology probably makes sense, but the switch routing configs will become more and more complex the more switches you add, hence my suggestion to move to dynamic routing.
Also, take the time now to plan out how your IP addressing scheme might look with more switches on more different sites. Allow a full /24 for every point-to-point link or bus, even if you don't use them. Since you are using RFC1918 addresses, you have space to burn (and when IPv6 comes, you'll have even more space to burn). Plan it so that things fit together logically (i recommend having some sort of fixed mapping between VLAN numbers and address ranges, e.g. VLAN 1010 == 10.10.0.0/16), and so that the addresses are easy to aggregate for routing efficiency (i.e. keep all of 10.10.0.0/16 on one site, all of 10.40.0.0/16 on another site, etc.).
Hope that all makes sense!