How do I deal with 'take backsies', particularly in Twilight Imperium?
You need, however annoying it may seem, to formalize a policy that deals not only with minor 'take back' adjustments, but with mistakes that were not noticed at the time. Before the next game, ask the group for a rule on which everyone can agree.

Some possibilities are: 


No changes after the dice have been rolled (for any reason). This has the advantages that it is clear, and if you forgot to take an action before you rolled the dice, you have nobody to blame but yourself. It has the disadvantage that the sort of player you describe may well miscount the numbers in a combat (accidentally or otherwise), announce the wrong odds, and roll the dice before anyone has a chance to object; does the result stand?
To avoid this (and other results of haste), no changes after the next player has rolled the dice. Even this is vulnerable to cheats (see The Bum's Rush in Monopoly), but it gives a reasonable period for checking.
Any changes you like, any time during your turn. It sounds as if you would hate this; but if you put it forward as a possibility, saying "That's what we were playing last time", the player in question may realize that other people can do it to him, so perhaps it's not such a good idea.
Whatever other players suggest in your particular game, including extra leeway for new players. Since this is a house rule, there is no 'correct' solution; but apparently two of you were playing by different house rules, so the table needs to agree. I would strongly advise against "a reasonable time" since that will be open to interpretation, and both sides of the argument will think the other is trying to cheat. Even "Five minutes after the action" is unlikely to work except in one of those games with an eggtimer or chess clock running at all times.