In my experience, it is extremely unusual in U.S. publishing style to include a period at the end a complete-sentence quotation that appears in the midst of a longer sentence of original text.
As Edwin Ashworth notes in a comment beneath the posted question, "It's standard to convert the logical period into a comma, indicating that the internal sentence ends but the matrix sentence doesn't." As an illustration of this point, consider this example from William Empson, in Argufying: Essays on Literature and Culture (1987), cited in Bryan Garner, Garner's Modern American Usage, third edition (2009):
The first word of reply of [Charles] Lamb (3 October) begins with the words, "Your letter was an inestimable treasure to me," but the next one grieves that Coleridge is not settling down to a serious course of life, and the third (24 October) questions the doctrines that Coleridge has preached: ...
A quick check of a reprinted version of Lamb's original letter in The New Monthly Magazine and Universal Register (August 1848) confirms that the sentence cited by Empson was indeed complete there:
MY DEAREST FRIEND,—Your letter was an inestimable treasure to me. It will comfort you to know, I know, that our prospects are somewhat brighter. ...
So we might have expected Empson to punctuate the quoted sentence with a period if his goal had been to be absolutely faithful to the punctuation used in the original letter—absent the U.S. convention of replacing an end period with a comma under such circumstances.
Another option is to use no end punctuation at all for the quoted complete sentence. An example of this kind appears in a guideline in The Chicago Manual of Style, sixteenth edition (2010) in the context of advice about whether to begin such quotations with a capital letter or a lowercase one:
13.14 Initial cap or lowercase—run-in quotations. When a quotation introduced midsentence forms a syntactical part of the sentence, it begins with a lowercase letter even if the original begins with a capital.
Ben Franklin admonishes us to "plough deep while sluggards sleep."
With another aphorism, he reminded his readers that "experience keeps a dear school, but fools will learn at no other"— an observation as true today as then.
When the sentence has a more remote syntactic relation to the rest of the sentence, the initial letter remains capitalized.
As Franklin advised, "Plough deep while sluggards sleep."
His aphorism "Experience keeps a dear school, but fools will learn at no other" is a cogent warning to people of all ages.
The instance most relevant to the posted question here is the last one, in which the aphorism about experience is embedded in the surrounding sentence without any opening or closing punctuation beyond the quotation marks themselves.
Applied to example noted in the posted question, the Empson and Chicago examples would seem to endorse (or at least countenance) the following three punctuation options:
1. In a disappointed and irritated tone, my mom suddenly said, "I always thought Sue would marry a Chinese person," right in front of him.
2. In a disappointed and irritated tone, my mom suddenly said, "I always thought Sue would marry a Chinese person"— right in front of him.
3. In a disappointed and irritated tone, my mom suddenly said, "I always thought Sue would marry a Chinese person" right in front of him.
