I gather that the 1% Tween is needed to prevent the protein from aggregating. If the SEC column was not equilibrated with Tween, the concentration of Tween that eluted with the protein may have been too low to prevent aggregation. To make sure the Tween concentration remains at 1%, put 1% Tween in the buffer used to equilibrate the column.
You may not need such a high concentration of Tween, however. It may be sufficient to use 0.1% or even less once the protein is largely purified, as long as it isn't too concentrated.
If the protein is incorporated into the Tween micelles, it will elute at a larger apparent molecular weight because of the size of the micelles. You may conclude that the protein is aggregated even if it isn't.
I centrifuged down the cloudy fractions from SEC and attempted to re-solubleize the resulting pellet with 1% Tween. (Barely any protein showed on SDS from the supernatant, leading me to post this question, perhaps I should have added Tween initially without pelleting). At 4 degrees Celsius rocking overnight, all of the flakes still did not go back into solution with the addition of this detergent.
I am going to retry the SEC purification step with Tween in the elution buffer like you suggested.