We are allowed to remove kosher tzitzis and not add them to another tallit, which I think is implied by your question, in order to make the tallit more beautiful (Baer Hetev sk.2, bringing the Levush). Shaarei Teshuva sk.2 says that this might be for example because the tzitzis are worn out or not as strong fabric as they could be. Mishnah Berurah sk.3 explains that this is because you are then enhancing the mitzvah of tzitzis, and that the prohibition is instead because of dishonoring the tallit; and also cites the Taz (sk.2) who says the same thing.
Those who add techelet to their tzitzis are doing so because they want to fully obey the mitzvah in the most appropriate way, even though we are allowed to fulfill it without techelet. So, they are trying to honour the tallit and tzitzis as much as possible, and therefore make it beautiful as techelet is beautiful. So, granting that we have the real techelet, it is permissible to remove kosher tzitzis to add ones with techelet.
Despite all this, the Piskei Teshuvos advises us to be makpid about avoiding removing tzitzis even if we want to add more beautiful ones. See Mor U'Ktziah of the Ya'avetz on Tur OC 15:
שאסור להתיר ציציות שלא להניחן בבגד אע"פ שלא יבטל הבגד ממצות ציצית, כגון שיעשה בו אחרים, ואפילו טובים מאלה
And the Shulchan Hatahor 15:1 of the Komarno rebbe:
אבל להניחם עד שיזדמן לו בגד אסור, אפילו שהם ישנים ובלוים, כל עוד שהם כשרים
Basically, unless the tzitzis are inappropriate/not kosher for the mitzvah, we should avoid adding more beautiful ones. Depending on whether we are makpid like this or more lenient like the plain reading of the sources further above, the question then resolves around how supporters of techelet view tzitzis without techelet.
Either we are lenient and then it is allowed at least to beautify the tzitzis. Or we are makpid, and then your situation is only allowed if you think tzitzis without techelet are actually inappropriate/not kosher for the mitzvah.