The theorem is, in fact, false even for finitely locally presentable $\infty$-categories, and indeed for the $\infty$-category of spaces, as has been known since Heller's paper
- On the Representability of Homotopy Functors, Journal of the London Mathematical Society (2) 23 (1981) pp. 551-562, doi:10.1112/jlms/s2-23.3.551.
Heller's counterexample relies on the existence of a group admitting a unique conjugacy class of injections from every group in any given small set; of course no such group exists handling all groups at once.
The basic problem is that the homotopy category of spaces has no small set of objects detecting isomorphisms, a critical ingredient in Brown's original work as well as all its more specialized successors. (You need a small set of types of cells to construct your representing complex out of.) Arguably, the whole historical focus of algebraic topology (and of the study of Brown representability in particular) on pointed connected spaces, which are $\infty$-categorically not the objects of interest at all, comes down to the need to have a set of objects, in this case the spheres, detecting isomorphisms.
As for your question, your formulation is more complex than needed: the only $\alpha$-filtered colimits you care about are weak ones in the homotopy category that you'll use to build up your representing cell complex out of generators, so the correct hypothesis, as it has always been since Brown, is just the preservation of coproducts and weak pushouts in the homotopy category. But you'll need the generators to appear $\alpha$-compact with respect to these weak colimits, and if $\alpha$ is uncountable then your homotopy category does not admit such constructions, see my preprint Detecting isomorphisms in the homotopy category with Christensen.
EDIT: Here is an explicit counterexample to the "only if" direction of the proposed theorem in a locally presentable $\infty$-category. (Note: there are no counterexamples to the "if" direction, but there are also no known examples of functors satisfying its hypotheses, even weakened to consider only sequences, with $\alpha$ uncountable.) Generally, it suffices to find an object $Z$ and an $\alpha$-indexed sequence $X_i$ together with a cocone under $(X_i)$ in the homotopy category with legs $f_i:X_i\to Z$ such that, for any choice of homotopies $H_{ij}:X_j\otimes \Delta^1\to Z$ between $f_i|_{X_j}$ and $f_j$, it is never possible to fill the induced maps $X_k\otimes \partial \Delta^1\to Z$ consisting of $H_{ik},H_{jk},$ and $H_{ij}|_{X_k}$. In this situation $(f_i)$ is a cocone in the homotopy category not inducing any map $\mathrm{colim} X_i\to Z$, which shows the functor represented by $Z$ does not send the $(\infty,1)$-colimit of the $\alpha$-indexed diagram $(X_i)$ to a weak colimit of sets.
The above framework is valid in any locally presentable $\infty$-category $\mathcal C$, and heuristically one should get counterexamples in essentially any such $\infty$-category and for essentially any choices of $Z$ and $(X_i)$. Franke's argument (introduction of On the Brown representability theorem for triangulated categories, Topology 40 (2001) pp.667-680, doi:10.1016/S0040-9383(99)00034-8) for this heuristic in the stable situation was as follows: There is a spectral sequence converging to $\mathrm{Hom}_{\mathcal C}(\mathrm{colim} X_i,Z)$ whose second page contains $\lim^q H^p(\mathrm{Hom}_{\mathcal C}(X_i,Z))$, and the functor represented by $Z$ will only send $\mathrm{colim} X_i$ to a weak limit of sets if this spectral sequence collapses. Unfortunately, nobody seems to know how to compute these derived limits well enough to show that what intuitively ought to happen actually does happen-there are isolated explicit examples of ordinal-indexed sequences of abelian groups with nontrivial derived limits, but I don't know how to cook one into this spectral sequence.
So the only place I know how to follow the recipe above is in unpointed spaces, and indeed in the homotopy category of groupoids, where I can compute stuff directly. Christensen and I analyze an appropriate space $Z$ in section 3.3 of our preprint. The idea is to start with $(X_i)$ itself, then construct $Z$ as a kind of unfinished homotopy colimit that has an incoherent family of homotopies between its canonical maps from the $X_i$. The utility of working with groupoids here is that one can combinatorially show that such a cocone, with intuitively shouldn't cohere into a map from $\mathrm{colim} X_i$, actually does not. In particular, this construction uses properties of the homotopy category of spaces that are totally orthogonal to the lack of generating cogroups, so should make one more confident that the expected phenomena would appear in the stable situation if we could wrangle the spectral sequence.
While the $\alpha$-filtered colimit idea does not work, Brown representability can be proved efficiently for these categories. Use the fact that an $\alpha$-presentable $\infty$-category is reflective in a finitely presentable one, and steal Brown representability from there--it's easy to prove that a reflective subcategory of a category on which every functor preserving coproducts and weak pushouts is representable enjoys the same properties. When the resulting finitely presentable $\infty$-category is generated by a set of compact cogroups, you win. Unfortunately, as discussed above, the homotopy categories of finitely presentable $\infty$-categories do not usually have Brown representability available to steal. There is a way out: the spheres do detect isomorphisms in the homotopy 2-category of the $\infty$-category of spaces, and from there it follows that the whole Brown representability story goes through perfectly for all $\alpha$-presentable $\infty$-categories--the only cost is that you have to take functors valued in groupoids, not in sets. Your mileage may vary on how "citeable" this is, but see 5.3.6 in my 2020 thesis, 2-categorical Brown representability and the relation between derivators and $\infty$-categories.
