Hi,
Did something similar in the past - although the timeframe was a bit different (at least 5 years between the alpha & gamma version) - we also skipped step 2 as at the moment we migrated to the gamma version we didn't know the url's of the alpha version anymore.
Changing url's is always a risky business from a SEO perspective and I hope you're certain that you will stick to the gamma version for the next couple of years.
To reply to your question - the method you propose looks good and is probably the best way to migrate the url's. I would advice to check & double check the 301's. Put the new structure on a test server (mytestdomain.com or on a subdomain like test.mydomain.com - block it for indexing). Activate the redirects on the test server. Use Screaming Frog (in list mode) with both the alpha & beta url's to check that these are properly redirected (you will need to update the url lists and replace the current domain by test.mydomain.com or mytestdomain.com for test purposes).
Also check do a full crawl on the test server (in crawl mode) to see that all the internal links have been updated (there shouldn't be any internal 301).
If this is ok - you can put the redirects & the new url structure on the production server. Doublecheck again with Screaming Frog to make sure that everything is ok (same tests as above put on the production environment. Monitor both Webmastertools and Analytics (make sure you can identify your 404 page in Analytics) to see if the number of 404 increases. If so, check the causes & correct them asap.
If tested like this, normally you shouldn't have a SEO impact.
Hope this helps,
Dirk