Take what the missing index view and DTA say with a pinch of salt, just dont apply anything it says at think it will work.

Your best off running DTA against a full workload, not just a single query, as DTA may advise on things which can impact other processes.

Run on a dev system, test it, test anything else which uses the tables you have just changed, then do the same on a test system, let the business use it, if they sign it off, then you can go to production