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If your MPLS vendor supports no dynamic routing, they why do you ask about BGP? Or, do they only support dynamic routing with BGP?
You can do equal cost multi-path with BGP (may require a hidden command to fully utilize).
You could do GRE tunnels across the MPLS cloud and dynamically route between them using your choice of a dynamic routing protocol.
Both your devices should support OER/PfR (may require a feature upgrade). OER/PfR will actually dynamically load balance.
SLA features should also be available on both your routers, those too might require a IOS feature upgrade.
Configuration examples might be found on Cisco's main web site.
