But we do not want to harm our existing SEO while we are doing this.
What you might be concerned here with is Cloaking - that is, serving different content to a user's browser than to search engine robots.
Providing that you use IP delivery, which is using the requestor's location to deliver content specifically written for that country, versus redirecting based on user agent, you should be fine as indicated here:
Many of the top 1000 sites, including sites like Amazon (amazon.com),
actively use IP delivery. None of these have been banned from search
engines as their intent is not deceptive.
You should probably also be aware of issues related to serving duplicate content between your international and country-specific website, should they contain the same content, which can be addressed with canonicalization and geotargeting your content.
To accomplish redirection with IP delivery, it would be wise to use a service that provides GeoIP databases which are routinely updated, such as MaxMind's GeoIP Country Database. You can implement that using server-side APIs in the following languages:
Many of these have easy to use open source modules too. For example, I use this PHP extension: Geo IP Location, as well as this Perl module: GEO-IP.
There's a free version of the database, GeoLite, which is updated once a month, and a more accurate paid version, GeoIP Country, that gets updated more frequently.