The canonical solution to this is to not rely on end user IP address, but instead use a Layer 7 (HTTP/HTTPS) load balancer with "Sticky Sessions" via a cookie.
Sticky sessions means the load balancer will always direct a given client to the same backend server. Via cookie means the load balancer (which is itself a fully capable HTTP device) inserts a cookie (which the load balancer creates and manages automagically) to remember which backend server a given HTTP connection should use.
The main downside to sticky sessions is that beckend server load can become somewhat un-even. The load balancer can only distribute load fairly when new connections are made, but given that existing connections may be long-lived in your scenario, then in some time periods load will not be distributed entirely fairly.
Just about every Layer 7 load balancer should be able to do this. On Unix/Linux, some common examples are nginx, HAProxy, Apsis Pound, Apache 2.2 with mod_proxy, and many more. On Windows 2008+ there is Microsoft Application Request Routing. As appliances, Coyote Point, loadbalancer.org, Kemp and Barracuda are common in the low-end space; and F5, Citrix NetScaler and others in high-end.
Willy Tarreau, the author of HAProxy, has a nice overview of load balancing techniques here.
About the DNS Round Robin:
Our intent was for the Round Robin DNS TTL value for our api.company.com (which we've set at 1 hour) to be honored by the downstream caching nameservers, OS caching layers, and client application layers.
It will not be. And DNS Round Robin isn't a good fit for load balancing. And if nothing else convinces you, keep in mind that modern clients may prefer one host over all others due to longest prefix match pinning, so if the mobile client changes IP address, it may choose to switch to another RR host.
Basically, it's okay to use DNS round robin as a coarse-grained load distribution, by pointing 2 or more RR records to highly available IP addresses, handled by real load balancers in active/passive or active/active HA. And if that's what you're doing, then you might as well serve those DNS RR records with long Time To Live values, since the associated IP addresses are highly available already.
You're right, the main disadvantage of using a low TTL is that you'll end up dealing with a higher query load than with higher TTL. That said, as long as your DNS host doesn't have any problems with handling the additional load, then go ahead. Honestly, if they had concerns about their ability to handle the load, they wouldn't let you set a very low TTL for your records (though many do tend to recommend that you set your TTL only as low as is necessary, which is a sound recommendation due to the user-experience point that ceejayoz made).
Additionally, I should mention that many ISPs DNS resolvers completely ignore the TTL setting and just set their own cache expiry on records, so for many cases, changing the TTL won't actually make any difference in the amount of queries that end up hitting the authoritative servers for your domain.
Best Answer
I'd recommend changing the TTL to something low a couple of days before you change DNS entries. That way, hopefully, when you change the dns the site will go live much quicker for most people.