3560s/3750s have enough memory to run several hundred BGP prefixes or more, if that meets your needs. That may be sufficient to run an IGP, or to advertise BGP to an upstream provider (and just take a default router via BGP in the downstream direction). Definitely won't take a full Internet table, but in many cases that's not necessary anyway.
Been there done that a few times. It works fine, and as always you just need to be mindful of the limitations of the platform in so far as memory and TCAM size. You will need the IPSERVICES featureset though, as BGP isn't supported on the IPBASE featureset.
Featurewise it's fairly much all OK, I can't remember anything missing in the implementation compared to much bigger switches and the config is exactly the same as all other IOS devices.
We are pleased to announce availability of Beta software for 16.6.3. 16.6.3 will be the second rebuild on the 16.6 release train targeted towards Catalyst 9500/9400/9300/3850/3650 switching platforms. We are looking for early feedback from custome...