The Sugia of hudcha or hutra is in Yoma 6b up to 8a and in Pessahim 77a. The issue there is the proportion in which uncleanliness is allowed for a korban tsibbur with fixed time. The gemara shows two opinions, the first, טומאת מת הותרה בציבור and the second, טומאת מת הודחה בציבור. This is not a radical controversy. We see two versions of the controversy in gemara. The hudcha in the second version is equivalent to the hutra in the first version. Additionally, the machloket seems to be in svara, not based on a controversial learning of verses. For instance, if there are Tahor (Ritually pure) and Tameh cohanim (Ritually impure), should the avoda be made by tahorim only? If tahor cohanim are not present in the bet av today, need we call cohanim from an other bet av?
There are similar dilemmas in pikuach nefesh doche Shabbat (regarding Shabbat the word is always "doche" but it is not relevant). Rishonim report questioning about in responsa of Raavad (the question is not the same in Rosh and in Ran) and responsa of Maharam Mirottenburg. The issue there is the choice between schita Shabbat and nevela for an ill man who needs to eat meat.
One of the rishonim, the Rashba, explains that the same controversy, seen regarding Tum'a is relevant for pikuach nefesh doche Shabbat. Last rishonim when they discussed the halacha didn't refer to the machloket regarding Tum'a.
The Rashba himself didn't say clearly that lenient Rishonim (the Maharam Mirottenburg) did pasken as the lenient opinion regarding Tum'a (i.e. they perhaps didn't link the issue of pikuach nefesh with tum'a hutra). But the Bet Yosef an Korban Nesanel understand this is the pshat in Rashba.
So, following this last lecture, Rambam rules hudcha in Shabbat and Tum'a, the Maharam Mirottenburg rules hutra in both.
I can add for fun only a personal chiddush. Tosfot in Yoma 6b, says that Rava probably follows the opinion that hutra, because, this is a Stam mishna (in his opinion). Someone taught me in mi yodeya that the Tosfot in Yoma is from the Maharam Mirottenburg. So, this indicates that Rava, who is basraa (chronologically late r than Rav Nachmann and Rav Sheshet) ruled the halacha following HUTRA. So, we can say that the Maharam Mirottenburg learned the Sugia as the Rashba but follows halachikally hutra and not hudcha.
The Shulchan Aruch OC 328, 14, rules hutra (Magen Abraham sk 9).
Generally we see several cases of asse doche lo taasse, which can show similar discussion s. EG if I can make a 4 canfot garment in wool or in linen. Can I lechatechilla make it in linen and use the asse to make shaatnez with the peril techelet.
A cohen that can make the funeral of his wife with non cohanim friends (מעשה ביוסף הכהן)...
So the answer is that the degree of leniency of chilul Shabbat is not necessarily linked with the words hutra and hudcha seen in the Sugia of Tum'a.