What do you call it when someone uses smart and intelligent but wrong statements to strengthen their argument?
This seems to fit your definition.
A 'specious argument' is one that is empty, false or incorrect.
All definitions are from Merriam-Webster. Words for this include shrewd
b : given to wily and artful ways or dealing
: full of wiles : crafty
2a : adept in the use of subtlety and cunning
1b : having surface plausibility or appeal : glossy
2a : characterized by subtlety or nimble wit : clever
1 : dexterous or crafty in the use of special resources (such as skill or knowledge) or in attaining an end
2 : characterized by wiliness and trickery
3 : prettily appealing : cute
4 : displaying keen insight
2a : clever in concealing one's aims or ends
Or clever, in a context where it implies dishonesty
2 : marked by wit or ingenuity
Or manipulative, from manipulate
b : to control or play upon by artful, unfair, or insidious means especially to one's own advantage
Casuistry can mean either a form of theological reasoning about ethics, or, more often:
2 : specious argument : rationalization
This is sometimes applied to specific, extant religious orders, but that would be considered offensive. Apologetics is also sometimes used neutrally as theological terms, but in other contexts, has a negative connotation of a lawyerly defense of the indefensible. An apologia can mean this too, but sometimes has a positive connotation.