The following is a very long comment and works in $1$-category theory.
I claim that you can characterize very well coreflective subcategories. My strategy could work even for reflective.
There is a biequivalence of $2$-categories $$ \text{Lex}^{op} \cong \text{Pres}. $$
- Lex is the category of small categories with finite limits, $1$-cells are functors preserving finite limits.
- Pres is the category of finitely presentable categories, $1$-cells are accessible right adjoints.
In a certain sense, the category Pres is the opposite of a locally presentable category. In fact, Lex is the category of algebras for the coKZ monad of free completion under finite limits (is this monad accessible?) on Cat. The category of algebras over an accessible monad defined on a locally presentable category is always locally presentable.
To conclude, as soon as one proves that Lex is locally presentable, in any way, one can derive a lot of results for Pres, just because it's its opposite category.