Can conservativity depend on the universe?

Probably not the best one can do, and what follows might be a bit 'overkill', but it answer the question about dependency on universe, and it is a nice argument.

Also if you know how the proof of the left transfer results I will use below works, it might give some idea on how to prove more general case of the result.

Theorem: Let $\kappa$ be an uncountable regular cardinal. Let $F: \mathcal{C} \to \mathcal{D}$ be a strongly $\kappa$-accessible left adjoint functor between locally $\kappa$-presentable categories. Then $F$ is conservative if and only if the restriction of $F$ to the full subcategory of $\kappa$-presentable objects is conservative.

(Here strongly $\kappa$-accessible = sends $\kappa$-presentable objects of $\mathcal{C}$ to $\kappa$-presentable objects of $\mathcal{D}$)

Proof: The weak factorization system (isomorphisms, all maps) on $\mathcal{D}$ is ($\kappa$-)combinatorial: it is generated by the empty set of maps. Hence the left transfered weak factorization system on $\mathcal{C}$ along $F$ exists and is also $\kappa$-combinatorial (I mean by this that is is cofibrantly generated by maps between $\kappa$-presentable objects). By definition, the left class of this weak factorization system is the set of maps such that $F(i)$ is an isomorphism. But if the restriction of $F$ to $\kappa$-presentable objects is conservative it means that all generators are isomorphisms hence, the left class only contains isomorphisms, so $F$ is conservative.

Note: the finer version of the left transfer theorem that I'm using which specify the presentability rank can be found as Theorem B.8.(4) in this paper of mine, but at the end of day it mostly follows from an analysis of the proof of the existence of left transfer available in the litterature (for e.g. in Makkai and Rosicky Cellular categories.)