What's the most common way to say "male chauvinist?"

The most common way to say "male chauvinist" is in fact male chauvinist.

But in your sentence, I would change it into an adjective:

Your ideology sounds a bit male chauvinistic to me.

Addendum (in response to comment)

It has been suggested in a comment that the word male could be omitted. Although the word chauvinist is (in my experience) most commonly heard in connection with male chauvinism, the meaning of the word is not actually restricted to that usage.

The following are definitions of chauvinism / chauvinist from three standard dictionaries:

chauvinism noun Chambers
derog an unreasonable belief, especially if aggressively expressed, in the superiority of one's own nation, sex, etc.
chauvinist noun, adj.
chauvinistic adj.
chauvinistically adverb
ETYMOLOGY: 19c: named after Nicolas Chauvin, a fanatically patriotic soldier under Napoleon.

chauvinist ODO
noun
a person displaying aggressive or exaggerated patriotism.
- a person displaying excessive or prejudiced support for their own cause, group, or sex:
- she wrote off all the local males as hopeless chauvinists
adjective
relating to or characteristic of a chauvinist:
- a chauvinist rejection of foreign interference

chauvinism M-W
1 : excessive or blind patriotism — compare jingoism
2 : undue partiality or attachment to a group or place to which one belongs or has belonged
3 : an attitude of superiority toward members of the opposite sex; also : behavior expressive of such an attitude
— chauvinist noun or adjective
— chauvinistic adjective
— chauvinistically adverb

It will be noted that the first definition in all three dictionaries relates to patriotism, rather than sexism. Indeed the word is derived from the name of a fanatically patriotic Napoleonic soldier.

Therefore, despite that fact that modern usage of the term seems primarily to relate to male sexism, it cannot always be assumed that that is the case. Indeed, there is nothing in the cited definitions to prevent the word being used to refer to female sexism. Having said that, if it is otherwise clear from the context that it is being used in reference, not only to sexism, but to male sexism, then, of course, the qualifier "male" could be omitted.

Consider androcentric if you are looking for a single-word synonym.

having or regarding man or the male sex as central or primary

The suggested alternative misogynist is not quite an accurate synonym. Whereas chauvinism implies belief in superiority or glory, misogynism specifically connotes hatred.

Androcentrism on the other hand carries the same connotations as chauvinism

Let me argue for a term you mentioned in your question: sexist.

ODO defines sexism as

prejudice, stereotyping, or discrimination, typically against women, on the basis of sex: sexism in language is an offensive reminder of the way the culture sees women

The term itself is arguably gender neutral, and there can be prejudice and other bad behavior toward men based on their sex. However, in practice, the term is overwhelmingly used to describe bad behavior toward women.

The term also describes a pattern of behavior that invariably views women as less important, less capable, less worthy and less self-directing than men. Again, there are arguments made that sexist conduct is meant to improve the lot of women (protect them, treat them with deference, etc.), but the views that espouse this are, at best, paternalistic.

Male chauvinistic is a pretty good term. I think sexist also fills the bill.

I would think the word 'misogynist' or 'misogynistic' to fit within the example sentence would be appropriate. Chauvinist can apply to nationalistic or racist feelings. Male chauvinist has specific, sexist connotation but in modern usage I feel you could just use the word 'chauvinistic' to convey the same meaning if you need it to be one word.