Rate Constant Units and Eyring Equation
Using Transition State Theory (TST) one has to be aware of at least some of the many flaws of it. And may also often lead to wrong assumptions. However, it is a wonderful tool to describe basic kinetics in a theoretical manner.

The most important/ fundamental assumptions are


separation of electron and nuclear movement (comparable to Born-Oppenheimer approximation)
(all possible) transition states (TS) are in equilibrium with the reactants  $$\ce{A + B &lt;=&gt; [AB]^{\ddagger}}\tag1$$
combined Boltzmann statistic for reactants and TS
All molecules that reach the transition state from the reaction coordinate of the reactants will leave the TS to the reaction coordinate of the product(s) - and vice versa 
\begin{align}
\ce{A + B &lt;=&gt; &amp;[AB]^{\ddagger}-&gt;P}\tag{2a}\\
\ce{P &lt;=&gt; &amp;[AB]^{\ddagger}-&gt; A + B}\tag{2b}
\end{align}
at the TS the transition vector can be separated from all other motions and hence may be treated classical as translation


In other words TST reduces a very complicated multidimensional problem to a one-dimensional translation. That implies that all treated molecules must be connected on a potential energy hypersurface with one vector. (Borrowing from the amazing wikipedia)



Hence reactions that are described with TST are of first order by default. One would have to make various adjustments to treat higher orders (and I would assume that not matching units would be of minor concern).