Yes, you need to purchase the proper IOS for the second router. The vendor should include that into your pricing.
When I procure networking equipment, I also include costs for the SmartNet (yearly reoccurring expense).
Lately, we've been ordering a lot of routers for MPLS sites. Each IOS that I order requires Advanced IP Services.
Your vendor should be able to help you with this question as well.
That said, if you have the credentials with your CCO account, it is possible to download the IOS and install it on your backup router. Or like you said, just copy the IOS to another router.
However, I don't think it's the right thing to do.
HTH
Mike