$X$ is paracompact Hausdorff iff $X\times Y$ is $T_4$ for all compact Hausdorff $Y$
For this theorem, the forward implication has a standard proof, while the reverse implication is generally proved by using Tamano's theorem, which uses compactifications.
However, I do not know much about compactifications. So, I would prefer if there was a proof for the reverse implication not using it. I've tried looking online, but to no avail. So, is there such a proof? Any help would be appreciated!
Lemma 2.5 (and surrounding results) do most of the work in this classic paper by Morita who proved this first. It used a compact test space of the ordinals $W(\omega_\alpha + 1)$ and does not use compactifications on a cursory glance over the proof. It's a generalisation (in a sense) of Dowker's result on countably paracompact spaces.
If the right hand side condition is fulfilled, for every cardinal $\mathfrak{m}$, $X \times [0,1]^{\mathfrak{m}}$ is $T_4$ which implies that $X$ is $\mathfrak{m}$-paracompact for all cardinals (this is in the paper) (and already Hausdorff trivially). So $X$ is paracompact Hausdorff.
This overview paper from 2002 by Noble might also interest you, as it is on similar questions. It also treats Noble's theorem that if $X$ is $T_1$ and $X^\kappa$ is normal for all $\kappa$, then $X$ is compact.