Hello,
Let's remember that the ASA was built as a High-Level Next Generation Firewall.
That does not mean it's not useful for routing but here we are talking about thousands of routes, I do not think there will be a performance issue on the FW because of that. I mean you have one of the greatest Cisco Firewalls (functionality and power speaking).
So if that's the case and you really want to do that you will need to implement either RIP,EIGRP,OSPF on the link and then do the redistribution on the routers.
Makes sense?
Regards,
Jcarvaja
CCIE 42930
Julio Carvajal
Senior Network Security and Core Specialist
CCIE #42930, 2xCCNP, JNCIP-SEC