Catholics must hold all the teachings of the church.

Yeah, I know that's such an excruciatingly obvious statement, so how is it that we can find so many Catholics who vocally deny or disagree with church teaching? Some ask why Catholics must believe what the Catholic church teaches. It's really simple. To do otherwise is to make oneself not a Catholic. ( Cf. C.C.C. 2089)*
Now, before you call "No True Scotsman!", remember that in some cases the argument that no member of a certain group does 'X' can be logically sound: If there are pre-requisites for being a member of a group , and someone doesn't meet them, they would not be a member of said group.
Now, a Catholic is one who is in full communion with the Catholic church. (Cf. C.C.C.)**
What does that mean?
Basically, someone who isn't separated formally or informally from the church by apostasy, schism, or heresy. But wait-
Oh Damn.
It's that "H" word.
The word no one wants to say, no one wants to use, no one even wants to thing about that particular word these days.
Heresy.
(S.1))Heresy is the obstinate post baptismal denial of some dogma of the Catholic church which must be believed with divine and catholic faith, (ie, binding on a person's conscience to believe) according to the Catechism, no. 2089.
(S.2)Heresy separates one from full communion with the church.
(S.3) Catholics are Christians who are in full communion with the Catholic church.
If all three propositions are true, then a person cannot be both a heretic ( A denier of some dogma of the church) and a Catholic. ( Who by definition of the terms "Catholic" and "Full Communion" does not deny any of the dogmatic teachings of the Catholic church.
The Code of Canon Law puts it :

"Can. 750 §1. A person must believe with divine and Catholic faith all those things contained in the word of God, written or handed on, that is, in the one deposit of faith entrusted to the Church, and at the same time proposed as divinely revealed either by the solemn magisterium of the Church or by its ordinary and universal magisterium which is manifested by the common adherence of the Christian faithful under the leadership of the sacred magisterium; therefore all are bound to avoid any doctrines whatsoever contrary to them.
§2. Each and every thing which is proposed definitively by the magisterium of the Church concerning the doctrine of faith and morals, that is, each and every thing which is required to safeguard reverently and to expound faithfully the same deposit of faith, is also to be firmly embraced and retained; therefore, one who rejects those propositions which are to be held definitively is opposed to the doctrine of the Catholic Church."

The Code of Canon law says of those who commit heresy:

Can. 1364 §1. Without prejudice to the prescript of can. 194, §1, n. 2, an apostate from the faith, a heretic, or a schismatic incurs a latae sententiae excommunication; in addition, a cleric can be punished with the penalties mentioned in can. 1336, §1, nn. 1, 2, and 3.

Were these not enough, The Catechism of the Council of Trent(Article IX) makes it quite clear:

"Heretics and schismatics are excluded from the Church, because they have separated from her and belong to her only as deserters belong to the army from which they have deserted....... Finally, excommunicated persons are not members of the Church, because they have been cut off by her sentence from the number of her children and belong not to her communion until they repent."

My reading of these is that people who deny dogmas are not in full communion with the Catholic church, and so, they aren't Catholic. Rather, as many modern canonists, at least since the SSPX near schism are using, they are in various degrees of impaired communion relative to their situation.
The two are mutually exclusive, and if the condition is lacking, the person isn't a member of the group.
According to Canon Law, denying the dogmatic teachings of the Catholic church makes you Definitively Opposed to the Catholic church. I don't see how, or even why you would claim membership of an organization that you are definitively opposed to. According to canon law, Catholics are bound to believe what the Catholic church teaches to be revealed by God, and to avoid doctrines that contradict them. Therefore, a Catholic cannot be free to deny what the Catholic church teaches to be revealed by God, and accept doctrines that contradict what she says. That contradicts. See also, for example, Donum Veritatis Nos. 22-23.

References:

Codex Iuris Canonici

Catechism of the Catholic Church

Catechism of the Council of Trent. (Roman Catechism)

Donum Veritatis

Citations:
*C.C.C. 2089: "schism is the refusal of submission to the Roman Pontiff or of communion with the members of the Church subject to him." Refusal to submit to the doctrines of faith or morals defined by him to be believe by obligation by all Catholics certainly applies.

**C.C.C. 837: "Fully incorporated into the society of the Church are those who, possessing the Spirit of Christ, accept all the means of salvation given to the Church together with her entire organization, and who - by the bonds constituted by the profession of faith, the sacraments, ecclesiastical government, and communion - are joined in the visible structure of the Church of Christ, who rules her through the Supreme Pontiff and the bishops. Even though incorporated into the Church, one who does not however persevere in charity is not saved. He remains indeed in the bosom of the Church, but 'in body' not 'in heart.'"