Re: Replacement meraki for core Catalyst 3750 switch
The closest option you could do would be to offload the BGP functionality onto an actual Cisco router. I like using the C1111-8P. Include an AppX licence to enable BGP.
It can easily flat line a Gigabit circuit. You could probably configure it as a "route server" as well as long as the other next-hop(s) are in the same subnet. Then the L3 forwarding is still done by the switch(es) while the BGP is done by the router(s).
The 3750 is so old and has a range of configurations and modules it is hard to suggest a direct replacement without knowing more. Options I like are:
MS250 - entry-level core switch that can also so OSPF and has 4 x SFP+ 10Gbe ports. Can take redundant power supplies.
MS350-24X - adds MGig ports, so you can also plug in 10GBaseT servers and SANs (as well as SFP+ 10Gbe).
MS425-16 - a pure 10Gbe SFP+ L3 switch. More common in small server farms or connecting switch blocks together (such as all the floor switch blocks in multi-story buildings).